ji^Tr    ^- 


-t> 


IN  MEMORIAM 
FLORIAN  CAJORI 


TKEATISE 


D^FFEEEI^^TIAL 


AND 


INTEGRAL  CALCULUS. 


BT 


PROFESSOR  THEODORE  STRONG,  LL.D., 

M 
JIkmbeb  of  the  "Ameeican  PniLosopnicAL  Society;"    "The  American  Academy  of 
Aeis  and  Sciences  ;"   and  ConroKATE  Membeb  or  "The  National 
Academy  of  Sciences,  U.  S.  A. 


NEW  YORK: 
0.  A.  ALVORD,  PRINTER,    15   YANDEWATER   STREET. 

1869. 


Entered  according  to  Act  of  Congress,  in  iLo  year  1869, 

Bt  THEODOKE  STEONG, 

In  the  Clerk's  Offic«  of  the  Dlatrict  Conrt  of  the  Uultcd  States  for  the  District  of 

New  Jersey. 


CONTENTS. 


DIFFEEENTIAL  CALCULUS. 


SECTION  I.  PAs» 

Definitions  and  First  Principles         .        .        -        -  1 

SECTION  II. 
Transcendental  Functions 49 

SECTION  ni. 
Yanisliing  Fractions  -        -        -        -        -        -        •  86 

SECTION  IV. 
Maxima  and  Minima 94 


VI  CONTENTS. 

SECTION  V. 
Tangents  and  Subtangeuts,  Normals  and  Subnormals        125 

SECTION  VI. 

Radii  of  Curvature,  Involutes  and  Evolutes       -        -        163 

SECTION  VII. 
Multiple  Points,  Cusps  or  Points  of  Regression         -        191 

SECTION  vni. 

Plane  and  Curve  Surfaces 205 

SECTION  IX. 

Curvature  of  Surfaces,  and  Curves  of  Double  Cur- 
vature -        - -        229 


mXEGEAL  CALCULUS. 


SECTION^  L  PAGB 

The  Integral  Calculus        -        -        -      .^        -        -        253 

SECTION  11. 

First  Principles  of  the  Calculus  of  Variations   -        -        316 

SECTION  III. 

Integration  of  Rational  Functions  of  Single  Variables, 

multiplied  by  the  Ditlerential  of  the  Variable     -        350 

SECTION  IV. 

Reductions  of  Binomial  Differentials  to  others  of  more 

simple  forms 377 

SECTION  V. 

Integration  of  Differential  Expressions  which  con^:  I 

two  or  more  Variables 439 


VIU  CONTENTS, 

SECTION  VI. 

Integration  of  Differential  Equations  of  the  first  order 

and  degree  between  two  Variables     -        -        -        454 

SECTION  VIL 

Integration  of  Differential  Equations  of  the  first  order 
and  higher  degrees,  and  the  Singular  Solutions  of 
Differential  Equations  between  two  Variables    -        484 

SECTION  vni. 

Integration  of  Differential  Equations  of  the  second 

and  higher  orders  between  two  Variables  -        -        511 

SECTION  IX. 

Integration  of  Differential  Equations  containing  three 

Variables 671 

SECTION  X. 

Partial  Differential  Equations    -        -        -         -        -        583 

Appendix  C02 


DIFFERENTIAL    CALCULUS. 

SECTION  L 

DEFINITIONS  AND  FIRST  PRINCIPLES. 

(1.)  In  the  Differential  Calculus,  numbers  or  quantities  are 
considered  as  being  constant  or  variable ;  tbose  whose  values 
do  not  change  during  any  investigation,  whether  they  are 
known  or  not,  being  called  constants ;  while  those  whose 
values  change,  or  are  conceived  to  be  altered,  are  called  vari- 
ahles.  Constants  are  generally  represented  by  the  first  letters 
of  the  alphabet,  and  variables  by  the  last  letters.  Thus  in 
anJG  ■\- h^  y  =  ax^  -\- hx  -[-  c  ;  a^h^  c  are  constants,  and  a?,  y  are 
variables. 

(2.)  Variables  that  are  entirely  arbitrary,  or  arbitrary  within 
certain  limits,  are  called  independent  variables  ;  while  those 
variables  whose  values  depend  on  the  values  of  one  or  more 
others  that  are  independent  of  them,  are  Q.?i^Qdi  functions  of 
the  variables,  on  whose  values  they  depend.  When  the  de 
pendence  of  a  variable  on  one  or  more  others  is  expressed  or 
given,  the  variable  is  called  an  exjMcit  functmi  of  the  va- 
riables on  whose  values  it  depends ;  but  if  the  manner  in 
which  a  variable  depends  on  one  or  more  others  is  neither 
expressed  nor  known,  and  is  to  be  found  from  the  solution 
of  one  or  more  equations  or  in  any  other  way,  the  variable 
is  called  an  implicit  function  of  the  variables  on  whose 
values  its  value  depends.     It  may  be  added,  that  a  variable 


2  DIFFERENTIATION   OF  ALGEBRAIC   EXPRESSIONS. 

which  is  expressed  in  variables  and  constants,  is  not  consid- 
ered as  being  a  function  of  the  constants.  Thus,  in  y  =  3jj 
+  7,  y  =  aa?  +  5,  y  is  an  explicit  function  of  a?,  and  x  is  an 
implicit  of  y ;  and  neither  y  nor  x  is  considered  as  being  a 
function  of  the  figures  3,  7,  or  of  the  constants  a,  h. 

To  signify  in  a  general  way,  that  any  variable,  as  y,  is  an 
explicit  function  of  another  variable,  as  a?,  we  write  them  in 
such  forms  as  y  =  F  {x\  y  =  f{x),  y  =  <p{x\y  =  4'  {x\  &c., 
either  of  which  is  read  by  saying  that  y  is  an  explicit 
function  of  x :  and  to  show  that  y  is  an  implicit  function 
of  a?,  we  use  such  forms  as  F  (a?,  y)  =  0,  f{x,  y)  =  0, 
^  (a?,  y)  =  0,  &c,  which  are  read  by  saying  thi^t  y  is  an  im- 
plicit function  of  x, 

(3.)  When  a  function  of  a  variable  and  the  variable  in- 
crease or  decrease  together,  the  function  is  sometimes  called 
an  increasing  function  /  but  if  the  function  increases  when 
the  variable  decreases,  or  the  reverse,  the  function  is  said  to 
be   a  decreasing  function.     Thus,  'm  y  z=  ax  +  h,  y  is  an 

increasing  function  of  x ;  and  in  ?/  =  — ,  y  is  a  decreasing  func- 

X 

tion  of  X. 

(4.)  If  X  represents  any  arbitrary  variable,  which  is  changed 
into  x' ;  then  x'—x,  the  difference  of  the  values  of  x  (found 
by  subtracting  the  first  from  the  second),  is  called  (in  the 
Differential  Calculus)-,  the  differential  of  x  (x  being  the  first 
value  of  the  variable),  and  is  expressed  by  dx,  by  writing  the 
small  letter  d  (the  first  letter  in  the  word  differential)  before 
or  to  the  left  of  the"  variable  x. 

If  X  stands  for  any  function  of  aj,  and  the  algebraic  sum 
of  aU  the  changes  in  the  valv^  of  X,  that  result  from  the  sep- 
arate variation  x'  —  x^=^  dx^  of  each  x  is  taken^  it  will  equal 
{what  is  called)  the  differential  of  X ;  which  is  expressed  by 


EXAMPLES   ON   DIFFERENTIATION'S.  3 

dX.,  as  in  the  case  of  x.  If  dX.  is  divided  by  dxj  the  qnotient 
is  called  the  dififerential  coefficient  of  dx  (the  differential 
of  the  independent  variable  a;),  since  it  is  the  coefficient  of 

dx  in  dX.  =  ~-  dx.     Because  constants  do  not  change  their 

(JiX 

values,  it  is  clear  that  tbe  differential  of  x  or  X  when  increased 
or  diminished  by  any  constant,  will  be  dx  or  6?X,  the  same 
as  before.  And  if  x  or  X  has  a  constant  factor  or  divisor, 
then  dx  or  <ZX,  when  multiplied  or  divided  by  the  constant, 
will  be  the  corresponding  differential. 

Thus,  if  X  :=  cc'  =  xx^  then  if  X^  represents  tlie  value  of 
X  when  either  x  (in  xx)  is  changed  into  x\  it  is  clear  that 
from  the  change  in  the  first  x  we  shall  get  X'=  x'x^  or  sub- 
tracting X  =  xx^  we  get  X'  —  X  =  x'x  —  ««  =  x  {x'  —  x) ; 
and  from  X  =  xx,  by  changing  the  second  x  into  x\  we  shall 
in  like  manner  get  X'  —  X  =  xx'  —  xx  ^=^  x  {x'  —  x) ; 
consequently,  from  the  addition  of  these  expressions,  we  get 
2  (X'-  X)  =  2x{x'  -x). 

Because  2x  (x'  —  x)  is  clearly  the  whole  change  that  can 
take  place  in  a?-  =  ,'»»,  ac3ording  to  the  preceding  principles  ; 
it  is  clear  (from  the  definition  of  the  differential  of  a  function 
of  a  variable)  that  for  2  (X'  —  X)  we  must  put  dX  the  differ- 
ential of  X,  and  since  x'  —  x  =  dx,  the  preceding  equation 
becomes  dX=:2xdx^  which  expresses  the  differential  of 
X  =:  x" ;  and  dividing  by  dx  (the  differential  of  tbe  inde- 
pendent variable  x),  we  have  -j-  =   2x,  for  tlie  differential 

coefficient  of  X  =  a?-.  If  X  equals  x^  or  a?*  or ...  .  .»",  n 
being  a  positive  whole  number,  we  shall,  in  like  manner, 
get  dX  =:  Sx'^dx  or  4:x'^dx  or  ...  .  n»"  -  ^dx,  for  their  differ- 

entials,  and  —7—=  Sx^  or  4:X^  or  ...  .  nx^~^  for  their  differ- 
dx 

ential  coefficients. 


4  EXAMPLES   (continued). 

Similarly,  if  X  =  aa?  +  J,  oraV  +  h\  ora'V  +  h'\  Sac,  we 
shall  get  c?X  =  adx  or  2a'xdx  or  Sa"x^dx,  &c.,  for  tlieir  dif- 

ferentials,  and  -t~  =  a  or  2«'aj  or  3a' V  or,  &c.,  for  their 

CtiC 

differential  coefficients;  and  generally,  m  being  a  positive 
integer,  the  differential  and  differential  coefficient  of  X  = 
{Ax"'  +  B)  -^  C,  will  be  expressed  by  ^ZX  =  7nAx"'-'^dx  -^  C, 

and  -7—  =  mAa?"*"^  -t-  C:  which  results  clearly  follow  from 

the  consideration  that  A,  B,  C,  do  not  change  their  values,  or 
that  their  differentials  equal  naught 

We  are  now  prepared  to  find  the  differential  of  a  variable, 
or  function  of  one  or  more  variables,  when  it  is  affected  by 
any  given  exponent ;  or,  as  is  sometimes  said,  we  are  pre- 
pared to  find  the  differential  of  any  given  power  or  root  of  a 
variable  or  function. 

n 

1.  Let  it  be  proposed  to  find  the  differential  of  X  =  x'" 
supposing  m  and  7i  to  be  positive  integers.     Since  the  equa- 

n 

tion  is  equivalent  to  X'"  =  a?",  which  is  the  same  as  (x")"^  =  a?** 
an  identical  equation  ;  by  taking  their  differentials  (according 
to  what  has  been  shown),  we  shall  clearly  havemX"*-WX  = 

n  (»i—  I)  n 

wa?"-^c^;  consequently,  since X'"-^=  a?"*  =  x"  '",  we  shall 
have  ^X  =  -  x"'"  dx^  as  required. 

i>  n 

2.  Let  X  =  aj  *",  or  Xa?"'^:  1,  be  proposed,  in  order  to  find 

n 

d^  the  differential  of  X  =  a?   "*  ,  supposing  as  before  m  and 

n 

f,  to  be  any  positive  integers.     Because  Xa;'"  =  1,  is  essen- 

n        n 

tially  the  same  as  the  identical  equation  x  "*  a?*"  =  1,  it  is  clear 
that  the  differential  of  Xaj"*  must  equal  naught,  since  the  dif- 
ferential of  its  equivalent,  1,  equals  naught 


DIFFERENTIAL   OF   A   POWER   OR   ROOT.  5 

It  is  clear  (from  tlie  nature  of  a  differential),  that  in  find- 

n 

ing  tlie  differential  of  Xx'"\  we  may  take  tlie  differential  of 
each  factor  regarding  the  other  as  constant,  and  add  the 
results  for  the  whole  differential ;  consequently  we  shall  have 

'X.dijo'''  -f  x'^'dX  =  0,  ordX.  =  —  ~X.x-\ix=  —  ^    ""     dx, 

771/  iiv 

as  required. 

3.  If  X  —  U     «  ±  aj    «  )    ''^  we  shall  clearly,  as  before, 

±£- 1 

have^X  =  ±  —-^l  a'l  ±  x^^]  x  \     dx.  for  its  differ- 

on  q\  J 

ential. 

4.  Hence,  the  differential  of  any  given  power,  or  root  of  a 
variable  or  function,  can  be  found  by  the  following 

RULE. 

Multijjly  the  jpower  or  root  hj  its  index,  siibtract  1  or 
unity  from  the  index,  in  the  jproduct  /  then,  multiply  the 
result  hy  the  differential  of  the  variable  or  function,  for  the 
reqidred  differential, 

EXAMPLES. 

1.  To  find  the  differentials  of  x^  and  {x'^y. 

Here  we  have  the  variable  x  raised  to  the  5th  power,  and 
the  function  x^  raised  to  the  ?ith  power,  the  indices  of  the 
powers  being  5  and  n  ;  consequently,  by  the  rule,  we  shall 
^s.MQhx^^dx  =  bx^dx  and 

n  {x^y-^dx"^  =  .^.^mn-m  ^  ,„^m-i^^  _  ^nnx'^^'-hlx 
for  their  differentials :  noticing,  that  the  second  differential 
is  manifestly  correct,  since  {oif'Y  =  a;"""*. 

2.  To  find  the  differentials  of   \/x  =  x^  and  \/x^  =  x^. 
Here  J  and  f  are  the  indices,  and  by  the  rule  we  shall  have 


6  EXAMPLES  (continued). 

ds/x  =  dJ'=\x^''^dx  =  \x-}dx  =  2l7aj^^  ^  %^^^'  ^^^ 

2    1  2     1 

dj^Q?  =  o  i—dx  =   Q    -—  <7a?;  for  the  differentials. 

3.  The  differentials  of  Ty'  and  6z%  are  42/flf?/and  Ss"*^/^; 
which  are  obtained  bj  multiplying  the  differentials  of  y^  and 
2",  by  their  coefficients  7  and  5,  as  we  clearly  ought  to  do. 

4.  The  differentials  of  ax"^  ±  h  and  -7  a?"  i  ^,  are 

max'^-'^dx  and    -j-x^'-hlx^ 

which  are  clearly  correct,  since  the  constants  connected  with 
the  variable  parts  by  ±,  must  clearly  disappear  when  the 
differentials  are  taken,  and  that  the  differentials  of  «a?"*  and 

-ya?"  must  evidently  be  a  and  -j  times  the  differentials  of  x^ 

and  x\ 

6.  The    differentials  of  2  \/{a^  4-  x^)  ^  2  (a^  -f-  serf,  and 

?v/(«2  4-  a;^)  =  I  {a'  +  a?^)^  are  2  {a'  +  x^y^xdx  ==      ^"^"^"^ 


and  ^  (a-  +  a?-)  ~^a?c?£c  = ^— ^ — - 

^  5(^^^  +  ar^)^ 

6.  The   differentials   of  (cr  -f-  a^)~^  and    (a*  —  aj^)~',  are 
_  4  (^2  ^  a!')-3izjc?ic  and  6  (a^  —  aj^)-'*^;^ 

7.  The  differentials  of  (a^  +  Sir^)-^  and  (a^  -  3aj^)~^,  are 
-  42  {a'  +  Sxy^xdx  =  -  r-^'^  and  — ^^. 

8.  The  differentials  of  {a"  +  x~Y'^  and  {a^—xr-y^,  are 

4a?"Wa?  4ar'r7aj  ,  4:X^dx 

and 


(a^  +  x-J  ~  {a'x'  +  If  («V  -  If 

9.  The  differentials  of  {2?/  +  'Sx-y  and  {2y-  -  Sx%  are 
(4/  +  6x')  {4.ydij  +  Qxdx)  and  2  (23/^  -  3a^)  (4yr/y  -  6a?^4 


PAETIAL  DIFFERENTIALS   AND   COEFFICIENTS.  7 

(5.)  If  X  is  a  function  of  any  number  of  variables  tbat  are 
independent  of  eacli  other,  it  is  customary  to  call  the  differ- 
ential of  X  taken  with  respect  to  any  one  of  the  independent 
variables^  a  partial  differential  of  X,  and  the  corresponding 
differential  coefficient  is  also  called  a  partial  differential  co- 
efficient I  and  the  algehraic  sum  of  all  the  partial  dfferen- 
tials  of  X,  is  called  its  total  differential. 

If  X  lias  two  or  more  terms  that  are  functions  of  tlie  same 
variable,  it  is  clear  tbat  we  may  find  the  differentials  of  such 
terms  as  before,  and  then  take  the  algebraic  sum  of  the  dif- 
ferentials for  the  differential  of  the  sum  of  such  terms. 

Thus,  if  X  is  a  function  of  x^  y,  2,  &c.,  we  shall  have  -y-  dx^ 

dx 

-j—dy^  -j-^^)  <^c.,  for  the  partial  differentials  of  X,  whose 

sum  p-ives  ^X  =  —7 — dx  -\ — -. — dy  ■\ j-dz  +,  &c. ;  for  the 

^  dx  dy    ^         dz         ^        ' 

complete  or  total  differential  of  X  ;  and  -y-,  -^-,  --y— ,  &c., 

are  the  partial  differential  coefficients.  And  if  we  have 
X  =  ^ax'  —  l)x-\-c^  by  taking  the  differentials  of  its  terms 
separately  we  shall  have  6axdx  and  —  hdx  for  the  partial  dif- 
ferentials, whose  sum  gives  <:ZX  =  6axdx  —  hdx  =  {6ax  —  h)dx 
for  the  complete  or  total  differential  of  the  proposed  expres- 
sion ;  and,  of  course,  —, —  =  Qax  ==  J  is  the  corresponding 
differential  coefficient. 

Eemarks. — 1.  If  X  is  a  function  of  a  single  variable,  its 
differential  coefficient  is  sometimes  indicated  by  writing  the 
capital  D  before  or  to  the  left  of  X :  thus,  DX  signifies 
that  the  differential  coefficient  of  X  is  to  be  taken :    as  in 


8  PARTIAL  DIFFERENTIALS  AND  COEFFICIENTS. 

D  (aar*  —hx  +  c)  =  Saa^  —  &,  called  the  first  derived  function 
of  oaj*  —  hx  +  c.     And  if  X  is  a  function  of  x^  y,  &c.,  the 

partial  differential  coefficients  -j-,  -i— ,  &c.,  are  sometimes 

expressed  by  the  forms  D^jX,  DyX,  &a 

2.  To  indicate  that  the  differential  of  a  compound  quan- 
tity is  to  be  taken,  we  put  it  under  a  vinculum  or  inclose  it 
in  a  parenthesis,  to  which  we  prefix  d.  or  d  (called  the  char- 
acteristic of  differentials),  and  when  the  differential  has  been 
found,  the  quantity  is  said  to  have  been  differentiated.  Thus 
d.(aP+y^  -—  az)  or  d  (s?  -{■  y^  —  az)  indicates  the  differential 
of  a^  +  y^  —  «2,  which  being  taken,  gives  d  {x^  -\-  y^  —  az)  = 
2xdx  +  2ydy  —  adz. 

To  make  what  has  been  done  more  evident,  take  the  fol- 
lowing 

EXAMPLES. 

1.  To  find  the  differential  and  differential  coefficients  of 
X  =  Za?-  5y  +  93\ 

Here  fZX  =  Qxdx  —  lOydy  +  27z'dz ;  and  ^  =  Qx, 

-T-  =  —  lOy,  and— i^ — =  27 z'.  are  the  differential  coefficients. 
dy  ^'         dz  ' 


2.  Perform  what  is  expressed  hj  d{  Vx^  —  2^^  +  az)  and 

^  (.^  _aj2  +  a;  -  8/  -  9y^  +  7). 

rr.1  xdx—2ydy         ,        , 

The  answers  are  -n—o — ^.^  +  adz,  and 
V(ar^-2?/-) 

^x^dx  —  2xdx  +  dx  —  Vly^dy  +  27y'dy  ; 

and  the  partial  differential  coefficients  are 

^  ^^  and3aj-2aj+l,-12y^+27y^ 


4/(ar-22^)'       i/(a^-2/) 


RULE   FOR  THE   DIFFERENTIAL   OF   A   PRODUCT.  9 

3.  Perform  wliat  is  indicated  by 

By  {x^  —  3/')  and  Djc^y  {ax^—y^  +  z). 

Ans.  —  15y*,  4:ax^,  and  —  5(/^ ;  when  Dx^y  is  used  to  indi- 
cate the  differential  coefficients  with  reference  to  x  and  y. 

4.  To  find  the  differential  of  the  product  of  any  number 
of  factors,  as  X,  Y,  Z,  &c.;  which  may  (if  required)  be  func- 
tions of  any  variables.  ■ 

Here  it  is  easy  to  perceive  (from  the  nature  of  differen- 
tials) that  d{XY)  ^  XdY  +  YdX, 

d  (XYZ)  =  TXdz  -\-XZdY  +  YZc^X,  &c., 

which  are  of  like  forms,  are  the  sought  answers. 

(6.)  It  clearly  follows  from  the  preceding  example,  that  the 
differential  of  a  product  can  be  found  by  the  following 

RULE. 

The  differential  of  the  product  of  any  number  of  variables 
or  functions,  eqxials  the  {algebraic)  sura  of  the  differentials, 
that  7'esidtfrom  the  differential,  of  each  factor  multiplied 
hy  the  product  of  all  the  remain  ing  factors, 

EXAMPLES. 

1.  The  differentials  of  xy  and  3-»'y,  are  xdy  +  ydx  and 
3  {ardy  +  lyxdx)  =  ?>di?dy  +  6yxdx. 

2.  The  differentials  of  xx^  and  x^t)^,  equal 

2x-dx  +  x^dx  =  Sx^dx,  and  4:X^x\lx  +  Sx^x'dx  =  7x^dx ; 
which  are  clearly  the  same  as  the  differentials  of  a^  and  a?^,  as 
they  ought  to  be. 

3.  The  differential  of  {x^  +  y-)  {x^  —  y%  is 

{a^  4-  2/')  (2a^^^'  -  2ydy)  +  {x^  -  f-)  {2xdx  +  2ydy) 
=  4  {x^dx  —  y'^dy). 


10  DIFFERENTIAL  OF  A   QUOTIENT: 

4.  The  differentials  of  y"  {a''  +  x'')  x  ^{a''-x-),  and  2A^ 


are 


x/Ix  ,,  ,  „,  Xf/X 


Va-  +  X'  X -■ -   +  {/(a^  —  x')    X 


%rMx 

and  5a?^ic*  <^ic  +  Wxdx  =  8ar\7a;. 

5.  The  differentials  of  <2a?yV  and -ar*y~^2~^,   are 

c 

ai^xy^zHz  ■\-  2xz^ydy  +  y-z^dx) 
and  ^  (-  Zx-y-'-z-^dz  -  2jrz-^if^dy  +  2y--z~''xdx). 

0 

X  tty 

6.  The  differentials  of  -  —xy-^  and  -,  =  a^'^y"^,  are 

tX  0  7.       17         V'^^'  —  ^dy 

y  ^      ^      J  2/2 

and  cZ~  =  2a??/  -^^.t?  —  3^t'-?/  '^dy. 

Bemark. — If  we  put  X  =  '-,    we    shall  have  Xy  =  a?, 

y 
whose    differential    gives    X^7y  +  y^ZX  =  Joj;     or,   since 

XXX 

X  =  -,  we  have  --  dy  -\-  yd  ~  =^  dx]  consequently,  we  shall 
J  J  " 

have    d—=  — 1, ~ ,  which    is    the    same    as   found 

y  r 

from  xy~^. 

(.7.)  It  follows  from  the  preceding  example  and  the  re- 
mark, that  the  differential  of  a  fraction  can  be  found  from 
the  following 

rule. 

IfaUiply  the  denom,imitor  hy  the  differential  of  the  nu- 
merator^ ajid  from  the  product  subtract  th-e  numerator  muU 


WITH  EXAMPLES.  11 

tiplied  hy  the  differential  of  the  denominator^  and  divide 
the  revfiainder  hy  tlie  square  of  the  denominator. 

EXAMPLES. 

1.    The  differentials   of  —  and  — ^r  are 

X  ar 

2xHx  —  x^dx      x^dx         ,           T    o?dx  —  23rdx  dx 
=^   =dx     and    —, =-^, 

which,  are  clearly  coiTect ;  since  the  form 

x^  ^    X  , 

- — ■  =  X,  and  —r  =  a?-\ 

X  XT 


- )  and  -— -,  are 
yl  ?/"* 


yi        y" 

"-^    'x\  _     ix^^-^ydx  —  xdy 


71 1        I  w  I         ■    —  / «/ 1         I  2 

ny'^x'^-'^dx  —  mx'^y^-^dy 

___ 

3.  Ilie  differentials  of  ^-=^  and4:i^;,  are^^^ 
T    .      (v^a?  —  xdy) 

4.  The  differentials  of  —  and  of =  —  ±1,  are 

X  X  X 


X  {ijdz  +  zdy)  —  yzdx  ^^.^     x 

or 


-3 and   -;;^^(«"±aj«)— (a"±a;")(^aj"  = 


a'^dai"'              na'^x'^-^dx  na"dx         ,«."        ,,        „. 
^-iT-  = ^—  =  -  -^.TT-  =  ^^  =  ^(«"^'^). 

5.    The  differentials  of and ,  are 7-- ^ 

a  -j-  X         a  —X  {a  -i-  xy 

,      adx 

and  -f r:7. 

\a  —  xy 


12        DIFFERENTIALS   OF  THE   SECOND,    FTC,    ORDERS: 


6.  The  differentials  of—jr-n ^  = r  and 

X  X 

.=  7T —11  are 


^{a'-ar)      (a^^a^f 


dx  x^dx  c^dx  ,         a^dx 

and 


7.  The  differential  of  — -t-v 5 is 

'^{a-  -\-  or)  -\-  x 

[V'(a^+i»-)  +  a?P      ~       a-  +  aj2  +  aJ4/(a^  +  a?)* 

dd'y 
noticing,  that  we  shall  in  like  manner  set  -5 s 7-5 sr 

for  the  differential  of  —,—= ^r . 

\/{ar  +  x-)  —  x 

(8.)  Supposing  X  to  be  a  function  of  x  alone,  taken  for 
the  independent  variable ;  then,  since  dX,  from  its  definition, 
equals  the  sum  of  all  the  changes  or  variations  in  X,  that 
result  from  the  separate  change  or  variation  x'—  x  =  dx  of 
each  x  in  X,  it  clearly  follows  that  the  differential  coefficient 

—=-  must  be  independent  of  dx;  and  that  a  double,  triple, 
(tx 

&a,  value  of  ^X  must  result  from  a  double,  triple,  &c.,  value 
of  dx,  and  so  on ;  and  it  is  clear  that  the  reverse  is  also  true. 
It  is  hence  evident  that  we  may,  according  to  custom,  sup- 

pose  dx  in  dX.  or  in  dX.  in  the  differential  coefficient  -j-  to 

(Xx 

be  unlimitedly  small,  and  that  when  x  is  the  independent 

variable,  dx  ought  to  be  regarded  as  constant  or  invariable, 

for  otherwise  x  must  be  regarded  as  a  function  of  a  variable, 

and  of  course  it  can  not  be  the  independent  variable. 


dX.  in  the  differential  coefficient  -y—  as  unlimitedly  small,  on 


WITH   THE   CORRESPONDING   COEFFICIENTS.  13 

It  is  further  evident  that  for  c?X  =  -^  dx^  "we  may,  if 

required,  write  dX.  =  -j-^  A,  and  regard  h  as  being  finite ; 

noticing,  that  it  will  generally  be  very  convenient  to  regard 

dX.  in  the  differential  coefficient 

account  of  the  minuteness  oidx. 

Calling  dX.  the  first  differential  of  X,  and  -j~  its  Jlrst 

dX 
differential  coefficient;  then,  if  tZX  or  -y—  contains  a?,  and 

CtdG 

we  take  the  differential  of  ^X,  supposing  dx  constant,  or  a?  to 
be  the  independent  variable,  we  shall  get  d  (<^X),  which  we 
shall  represent  by  c^'^X,  and  it  will  be  what  is  called  the 

second  difiexential  of  X  ;  and  d  -= — \-  dx=^  -j^ .  which  is 

ax  dx' 

the  same  as  dX  ~  dx^  =^  -p, ,  will  be  what  is  called  the  sec- 
ond differential  coefficient  of  X. 

d^X 
In  like  manner,  since  -j-;  is  clearly  independent  of  dx,  if 

cox 

it  contains  x,  we  shall,  as  before,  get  d  (d^X)  =  d^X  for  the 

third  differential  of  X,  and      \  ^     -^  dx  =  -^ ,  which  is 

dX 
clearly  the  same  as  d^X  -i-  dx^=  -^--3,  will  clearly  be  what  is 

6?  fly 

called  the  third  differential  coefficient  of  X. 

And  we  may  in  the  same  way  proceed  to  find  ^^X,  d^X . . 

^"X,  which  are  called  the  fourth,  fifth to  the  nth  differ- 

^^X 
ential;  and  the  corresponding  differential  coefficients,  -^-r, 

dx^ 

d^X        d'^X  — 

-^ -^-^.     Thus,  from  X  =  a?"  we  get  dX  =  nx''-^da!j 


14        IMPORTANT  RULE  IN  DEVELOPMENT. 

d'X  =  n  {n-l)  a;"-=<7.r',  d^X  =  n  (n-l)  {n-2)  x'^da?,  &c, 

dX 
for  the  first,  second,  third,  &c.,  differentials;  and— p-=?id?"-\ 

ax 

^  =  n  (71-1)  af^-',  ^  =  ^  (^^-1)  (^*  -  2)  a?"-',  &c,  will  bo 

the  corresponding  diifferential  coefficients  of  x\ 

If  we  put  x'—x=h,  or  x'  =  X  -{-  h,  and  change  a?  in  X 

into  x' ;  then,  if  the  resulting  value  of  X  is  expressed  by  X', 

it  is  manifest  that  X'  is  a  function  of  x'  or  its  equal  x  +  h. 
If  X'  is  developed  into  a  series  arranged  according  to  the 

ascending  powers  of  A,  it  is  evident  (from  what  has  been 

done)  that  X  and  -r-  A  will  be  the  first  and  second  terms  of 
^  ax 

the  series,  so  that  we  shall  have  X'  =  X  +  -^-A  +,  &c. 

Since  x  may  stand  for  any  variable,  and  X  for  any  func- 
tion of  it,  it  results  from  the  preceding  equation,  that  we  can 
find  the  first  difierential  of  the  function  by  the  following 

RULE. 

Change  x  in  the  function  into  a?  +  A,  and  develop  the 
resulting  function  into  a  series  arranged  according  to  the 
ascending  powers  of  A ;  then  the  coefficient  of  h  (the  simple 

power  of  h)  in  the  development,  will  equal  -;-  the  first  dif- 
ferential coefficient,  which  multiplied  by  dx  (supposed  unlim- 
itedly  small)  gives  -y-  dx ;  which  is  the  first  differential  of 

the  function  X, 

Thus,  if  we  put  X  =  ar',  we  get 

X'  =  {x  +  hf  =  0^  +  Sx'h  +  Sa-A^  +  h' ; 
consequently,  Sjt  =  the  first  difierential  coefficient,  and  of 
course  da^  =  Sardx  is  the  first  differential 


EXAMPLES   IN   HIGHEK   ORDERS   OF   DIFFERENTIALS.     15 

m 

Similarly,  from  X  =  ^' "  we  get 

!L^         HI         m     ^-1 
X!  =  {x  -\-  h)"  =  X''  +    —x''     h  +,  &c., 

as  is  clear  from  tlie  Binomial  Theorem. 

rn,   ——1 
Hence,  since  —a?''        is    the    coefficient    of   the    simple 

power  of  A,  in  the  expansion ;  it  follows  that  we  shall  have 

dx"^  ■=  — x'^     dx  for  the  differential. 

71 

m 

Kemark. — The  same  differentials  of  x^  and  a?",  can  be  ob- 
tained immediately  from  the  rule  at  page  5. 

(9.)  We  will  now  show  how  to  find  the  remaining  terms  of 

J'V' 

the  series,  X'  =  X  H — ^  A  + ,  &c. 
dx 

Thus,   by   taking  the  differential  of   j— A,   supposing  x 

alone  to  vary,  we  have,  according  to  the  principles  hereto- 

fore  given,  ~r^hh  =  jtt^^^i  ^^^  twice  the  third  term  of  the 

series.  For  any  term  in  -^  hh,  that  results  from  the  multi- 
plication of  terms  containing  A  and  A  taken  in  any  order, 
will  clearly  result  from  the  same  terms  when  A  and  A  are 
interchanged,  as  is  manifest  from  the  manner  of  obtaining 

-- TT  /<  A  ;  consequently,  —^ —  :r-^  is  the  third  term  of  the  series. 

ClX^  CCX    x.Ai 

Similarly,  from  -^  ^i^  we  get  ~j  t-^  for  thrice  the  fourth 

term  of  the  series. 

<^^X  A^A 
For  it  is  plain  that  any  term  in  -^-j-  z-^,  that  results  from 

the  multiplication  of  a  term  that  contains  A^  by  another  that 


16  Taylor's  theorem. 

contains  A,  will  equally  result  in  two  other  ways,  since  h^  can 
be  formed  in  two  other  ways,  by  combining  each  h  in  the  first 

h?  with  the  remaining  h ;  consequently,  -j-^-  ■Tn'o  ^  *^®  fourth 

term  of  the  series. 

It  is  hence  easy  to  perceive  that  -77^  is  the  fifth 

term  of  the  series,  and  so  on. 

For  a  more  full  explanation  of  the  principles  used  in  find- 
ing the  preceding  terms,  we  shall  refer  to  the  solution  of 
Example  16,  at  p.  56  of  my  Algebra,  and  for  the  common 
way  of  finding  them,  see  p.  252  (49.),  of  the  same  work  : 
observing,  that  this  method  is  altogether  more  complicated 
than  the  preceding. 

Hence,  collecting  the  terms,  we  shall  have 

whose  law  of  continuation  is  manifest :  noticing,  that  h  may 
be  positive  or  negative,  according  to  the  nature  of  the  case. 

Because  X'  is  the  same  function  of  a;  +  A  that  X  is  of  a?, 
it  follows,  if  we  represent  X  by  f{x)  =  any  function  of  a?, 
that  X'  will  become  a  similar  function  of  x  +  h,  represented 
byy(a;  +  A) ;  consequently,  the  series  (a)  becomes 

/(.+A)=/(.)  +  -^A  +  -^_  +  .^__+,&c..(«0. 

{a)  and  {a')  are  different  forms  of  what  is  called  Taylor'^s 
Theorem^  which  is  always  true  when  x  and  A  are  undeter- 
mined quantities,  or  when  the  series  does  not  contain  any 
fractional  or  negative  powers  of  A.  When  particular  values 
are  assigned  to  x  and  A,  the  series  afe  also  true,  provided  that 
no  term  becomes  infinite ;  but  if  one  or  more  terms  become 


maclaurin's  theorem:.  17 

infinite,  the  series  are  true  no  further  than  to  their  first  in 
finite  terms,  exclusively. 

If  we  represent  the  particular  values  of  X,  —7-,  ~rj^  &c., 

that  correspond  to  a?  =  0,  by  (X),   ("7— ),  \~t^\  ^^  ?  then, 

if  we  change  h  into  a?,  and  represent  the  corresponding  value 
of  X'  by  X,  {a)  will  become 

^     ,^,      /dX\         /cPX\  x^        /d'X\     ^ 

which  is  called  Maclaurin's  Theorem ;  in  which  x  may  be 
positive  or  negative,  according  to  the  nature  of  the  case. 
Because  X  is  supposed  to  be  a  finite  function  of  a?,  it  clearly 
follows,  if  (h)  gives  an  infinite  value  to  any  term  of  X,  that 
{]))  is  not  applicable  to  the  expansion  of  X. 

To  perceive  the  uses  of  Taylor's  and  Maclaurin's  Theo- 
rems, take  the  following 

EXAMPLES. 

1.  To  expand  {x  +  h)\  by  Taylor's  Theorem. 

Here  {x  +  hf  and  x"^  must  be  used  for  X'  and  X ;  which 
.      ^        ,  dX.      .  .  d^X      ^^  ,  d'X      ^,     d'X      ^, 

and  thence  (a)  becomes 

{x  +  Kf  ^x'-\-  4a?Vi  +  6a?Vi=  +  ^xh'  +  M 

2.  To  expand  {x  +  A)",  according  to  the  ascending  powers 
of  A,  by  Taylor's  Theorem. 

Here  X'  =  {x-Y  h)%  X  =  x-,'^k  =  nx--% 

^X   h^  _n{n-l) 

dx"  1.2  ~     1.2  ' 


18  THESE  THEOREMS   ILLUSTRATED. 

^  .  J!_  -  n(n-l)(n-2)  . 

dx'  1.2.3"  1.2.3  '^^•' 

consequently,  from  (a)  we  have  {x  +  A)"  = 

3.  To  expand  X  =  (a  +  a?/,  according  to  the  ascending 
powei-s  of  X,  hy  Maclaurin's  Theorem. 

Here  X  =  (a  +  xf  gives 

consequently,  jutting  a?  =  0  in  these,  we  get 

.,..        3     /dX\      _  ,     /c£^X\       .        ^X      . 

and  thence,  from  (A)  we  have 

{a+  xf  =za'  +  3a'x  +  Sax"  +  x\ 
as  required. 

4.  To  expand  ^ ^  (1  +  x)-^  according  to  the  ascend^ 

X  -f-  X 

ing  powers  of  x,  by  Maclaurin's  Theorem. 
Since  X  =  (1  +  x)~^,  we  have 

^-  =  -  (1  +  .^)  -,  ^=2a+x)-»,  -_  =  _2  x3(l+»)-S 

and  so  on ;   consequently,  by  putting  .o?  =  0   in  these,  we 
have 

and  so  on.     From  the  substitution  of  the  preceding  values 
in  (^v),  we  get 

=  l  —  x-\-o^  —  x^-{-,  &c., 


1+x 
for  the  required  expansion. 


EXAMPLES   (continued).  19 

5.  To  expand  —7- -,  according  to  the  ascending  powers 

iC  (  J.    Xj 

of  a?,  by  Maclaurin's  Theorem. 

Because  x  =  0  reduces  —zr- — r  to  7-  =  infinity,  it  would 
w(l  —x)        0 

seem  that  (h)  is  not  applicable  to  the  question.     Nevertheless, 

since  (b)  gives 

1 


1-x 

we  shall  of  course  have 
1 


1  -\-  X  i-  x^  +  x^  +,  &c., 


=  a?-^  +  l  +  a?  +  a?^  +  aj^+,  &c., 


x{l  —  x) 
as  required. 

6.  To  expand  X  =  -  =  aa?^'"  into   a  series,   an-anged 

X 

according  to  the   ascending  powers   of  x^  bj  Maclaurin's 
Theorem. 

Since  -^  —  —  ftax-''-^  — ^  =  n  (n  +  1)  »-"-",  &;c.,  71 
dx  dx- 

being  positive ;  and  since  these  are  infinite  when  a?  =  0,  it  is 

clear  that  (b)  is  not  applicable  to  the  question. 

7.  To  expand  X'  bj  (a)  on  Taylor  s  Theorem. 
Here,   by  {a)  we  have 

X'  z=  {x  -\-Jif  +  (a;  +  A  -  a)-^  andX  =  a;^  +  (a;  -  ay  ; 
and  putting  x—-a^  these  equations  become 

X^  z=  («  +  Kf  +  A-^-  =  (a  +  A)3  +  ^,  and  X  ^a'  ^-  ~. 

It  is  hence  clear  that  {a)  is  not  applicable  to  the  question 
any  further  than  to  the  expansion  of  {a  +  Kf. 

Remarks.: — It  is  manifest  from  what  has  been  done,  that 
X'  =  {^a  ■\-lif  4-  A-'  =  a«  +  h-^  ■\-Za-h  +  ZalC-  -f-A' 


20  NOTICING  SOME   FAILING  CASES 

is  the  true  expansion  of  the  proposed  expression,  when 
a?  =  <i.  And  it  is  clear  that  the  existence  of  A  ~  *  in  the 
expansion,  is  the  true  reason  why  it  can  not  be  found  by 
Taylor's  Theorem ;  since  it  (the  Theorem)  evidently  sup- 
poses the  indices  of  h  not  only  to  be  integral,  but  to  be  pos- 
itive. 

8.  To  expand  (a?  +  Kf  +  \{x  +  A)^  —  a^]*  according  to  tho 
ascending  powers  of  A,  when  x=a^  by  Taylor's  Theorem. 

Here  we  have 

X  =  a?*  4-  (^  -  a^)^, 


dx 


Because  a?^  —  a^  with  a  fractional  index  enters  the  denomi- 

nator  of  one  of  the  fractional  terms  of  the  value  of  -^i^i  ^^ 

clearly  follows  that  x^  —  a^  with  fractional  indices  must  enter 

as  divisors,  in   fractionid  terms  of  the  values  of-,-^, -r^, 

dx^'  dx^^ 

and  so  on ;  and  it  is  manifest  that  like  conclusions  must  hold 

good  in  all  similar  cases. 

The  substitution  of  the  values  of  X,  -^-,  «&;c.,in  («),  gives 

{x  +  hj  +  [(aj  +  Kf  -a^]  ^=^x'  ^  (x^  -  «=)'  + 

[2x+Sx{x^-a^)^]  A4-(2  -^S{x^-cr)^  +71^-^)  ^+ M. 
^  {or  — a-y'  J-^ 

for  the  expansion ;  when  no  particular  value  is  assigned  to  x. 


OF  Taylor's  theokem.  21 

When  x  —  a^  tlie  expansion  is  easily  reduced  to 

{x  +  Iif-\-[(x+Jif  -  a']"  =a'  +  2ah  +  h'  +   ^^f^, wliich 

is  true  in  its  first  three  terms,  but  not  in  its  fourth  term, 
which  is  infinite. 

To  find  the  true  expansion  when  x  ^  a^  we  put  a  for  x  in 
(a;  \  }if-\-\{x  +  Kf—a?)^^  and  thence,  by  a  simple  reduction, 
get  {a  +  hf  +  i^ah  +  hrf  ;  which  expanded  by  the  Bino- 
mial Theorem,  or  in  any  other  way,  gives 

{a  +  hf  +  {2ah  +  h^f 

=  ^^  +  2ah  +  {2ahf  +  A^  +  8  |/|  A^  +  ,  &c., 
for  the  true  expansion. 

Kemarks. — From  this  and  the  preceding  example,  we  per- 
ceive how  the  correct  expansion  may  be  found  when  h  is 
affected  with  a  fractional  or  negative  exponent,  in  conse- 
quence of  assigning  a  particular  value  to  x ;  which  are  gen- 
erally said  to  fall  under  the  failing  cases  in  the  applications 
of  Taylor's  Theorem :  noticing  that  the  Theorem  is  clearly 
not  applicable  in  such  cases,  because  it  supposes  the  indices 
of  h  in  the  expansion,  to  be  positive  integers. 

(10.)  If  X  is  a  function  of  any  number  of  variables,  as 
x^  y,  5,  &c.;  then,  if  a?,  y^  &c.,  are  changed  successively  into 
i»  +  A,  y  +  2*,  s  +  ^,  &c.,  the  resulting  values  of  X  may  be 
expanded  into  series  arranged  according  to  the  ascending 
positive  and  integral  powers  of  A,  ^,  h^  &;c.,  and  their  prod- 
ucts, provided  no  particular  values  are  assigned  to  x,  y,  ^,  &c. 

For  let  X'  denote  the  value  of  X  that  results  from  changing 
x  into  a?  H-  A ;  then  {a)  p.  16,  will  be  the  expansion.     In  like 

manner,  if  y  in  X,  -r-  A,     —^    —^  ,  &c.,  is   changed   into 


22 


Taylor's  theorem  generalized. 


y  -\-tj  and  the  resulting  values  of  X,   -y-  A,  &c.,  are  ex- 

panded  according  to  the  ascending  powers  of  /,  as  in  (a) ; 
then,  if  X''  stands  for  the  resulting  value  of  X',  when  y  is 
changed  into  y  +  i,  we  shall  have 


X-  ==:X   + 


^X    ,       ^X   h'       d'X 


/i' 


dx 

dX 

dy 


dx"  1.2 
««^X    hi 


d3^  1.2.8 


+  ,&c. 


dxdy  i  1      dardy  1.2     1 


d'X   i' 
dy'  1.2 


^  ^/.iv/e/^^  1.2  +'^''' 


+ 


^^X 


+,&c. 


(«") 


dy"  1.2.3 

If  we  had  at  first  changed  y  in  X  into  y  +  i,  and  then 
expanded  as  in  (a),  by  aiTanging  the  tenns  according  to  the 
ascending  powers  of  ^ ;  then,  by  changing  x  into  x  -\-  h  in 
the  terms 

^     ^X    .     c^X        7*=       , 
^'  ^''    -^    T2'  ^"•' 

we  should  in  a  similar  way  have  obtained  the  same  expan- 
sion under  another  form.     Thus,  we  have 


^,,_  dX        c^X   i'    ^d^X     {^       ^    . 


dX  -  ,    d'X    .,       (fX     *^    ,       , 
dx  dydv  dydx  1.2         ' 


cT-X    J^_ 
"^  c/a;"^  1.2  "^  c^^^.^'2 


*     — 7^    +  ,  &C. 


(a-) 


1.2 


d^    1.2.3 


+  ,&c. 


Because  the  preceding  values  of  X'^  ought  clearly  to  be 
identical,  we  must  have  the  equations 


DIFFERENTIATION   GENERALIZED.  23 

d'X  d'X        d'X  d'X        d'X  d'X 


dxdy       dydx^    d;i?dy      dydx  '  dxdy^      dy'dx^ 

dx'''df'   ""  dy^x"'' 

which  show  the  differentials  indicated  in  the  first  and  second 
members  of  these  equations  to  be  identical,  as  they  clearly 
ought  to  be,  from  the  nature  of  the  differential  calculus. 

Changing  z  into  z-\-k  either  in  (a")  or  {a"%  and  then  pro- 
ceeding as  before,  we  shall  get  the  expansion  that  results 
from  changing  .t,  ?/,  z  into  a  +  h^  y  +  h  a^d  z  +  h\  and  we 
may  proceed  in  like  manner  with  reference  to  any  number 
of  independent  variables  that  may  be  contained  in  X ;  and 
the  final  result  will  clearly  be  a  generalization  of  (a)  or 
Taylor's  Theorem. 

If  for  h  we  put  dx,  (a)  becomes 

X'  =  x  +  ^  +  g+j^-+,&c {ary, 

which,  according  to  the  preceding  generalization  of  (cr),  is 
true  when  X  is  a  function  of  any  number  of  independent 
variables,  and  that  whether  the  differentials  are  taken  rela- 
tively to  all  the  variables,  or  not 

EXAMPLES. 

1.  Given  X  =  a^,  to  find  its  differentials: 
Here, 

X  =  a^,  and  X'  =  (a?  +  dxf  =  a?^  +  Zx-dx  -f  Zxdx-  -f  dd(?; 
consequently,  since 


X'  =  X  +  dX  +  g+,&c., 


we  have 


24  EXAMPLES. 

"wLicli  must  clearly  be  an  identical  equation. 

Hence,  from  a  comparison  of  like  terms,  we  get 
X  =  a^,  dX.  =  Bardx,  d'X  =  1.2.3xdx  =  Qxdx,  d^X  =  6dx\ 
as  required. 

2.  To  find  the  differentials  of  X  =  aj*.' 

Since  X'=Xh-(£X+   f=+,&c., 

.         -  .i         \       \dx      \  d:&        1    daf        . 

=  i,  +  d.f=.i +  ----- -  +  --^-,&c.;     ■ 

consequently,  from  a  comparison  of  like  terms,  we  have 

x  =  x\  dyL  =  %  ^x  =  -^„^^x  =  ^^, 

2x^  4»«  8.C* 

and  so  on,  indefinitely. 

3.  To  find  the  differentials  of  X  =  xy. 

Since  X:=^'X  +  dX-\-'^  =  {x-\-dx).{7/  +  dy) 

=  xy  -\-  ydx  +  xdy  -f  dxdy^ 
an  identical  equation :  its  like  terms  equated,  give 

X  =  xy^  dX.  =  ydx  -\-  xdy^  and  drX  =  ^dxdy^ 
as  required. 

4  To  find  the  differentials  of  X  =  xy\ 

YiomX'=X  +  dX+^  +  ^^  =  {x+dx).{y  +  dyy 

=  a^  4-  2xydy  -f  yMx  +  xdy^  +  2ydxdy  +  dxdy% 
we  readily  get 

X  =  xif,  dX.  =  2xydy  +  yhlx^ 

d'X  =  2xdy^  +  ^ydxdy,  d'X  =  Uxdy\ 


MACLAUKIN  S  THEOREM. 


25 


Again,  if  we  put  x  and  y  in  {a")  each  equal  to  naught, 
and  represent  the  corresponding  values  of 

then  changing  li  and  i  into  x  and  y,  {a"^  becomes 


+ 


(|h/+r^^ 


)y  +  l;i^j^+'*°- 


(JO; 


which  is  Maclaurin's  Theorem  extended  to  the  expansion  of 
a  function  of  two  independent  variables :  and  it  is  easy  to 
perceive  how  Maclaurin's  Theorem  may  be  extended  to  the 
expansion  of  a  function  of  any  number  of  independent 
variables. 

To  illustrate  (5'),  we  will  apply  it  to  the  expansion  of 
X  =  (aa?  +  ly)\ 


Here 


and 


c^X      _    ,       ,  X  N  ^'X 


2a^g  =  0,&c. 


f  =  2^(-  +  ^^)'$-2^^f  =  <>'^-' 


dxdy  '  dydx 


2ab. 


Putting  a?  =  0  and  y  =  0  in  X  and  these  values,  we  get 

-.   (f)=«.(f)=-.f=».-^ 

\dxdyl  ~        * 


26  EXAMPLES. 

Substituting  these  values  in  {h'\  we  get 

X  =  {ax  +  hyf  =  y-^-  4-  2ahxy  +  y-|-  =aV+2a5a?y  +  %=; 

whicli  is  evidently  correct 

Kemarks. — 1.  li  cux  +  hy  is  eliminated  from  the  equations 
-^  =  2a  {ax  +  %)  and  ~^=2h{ax  +  hy\ 

dX.      jdX      - 

we  get  a—, 6— -=0, 

ay         dx         ^ 

which  is  called  an  equation  of  partial  differences  ;  but,  since 
-J-  and  -7-  are  differential  coefficients,  it  is  clearly  more  cor- 
rect to  call  it  an  equation  of  partial  differential  coefficients. 
2.  If  X  =  f{ax  +  hy)  =  some  function  of  oa?  +  hy^  it  is 

easy  to  perceive  that  -^  and  -j-   will  be  of  the  forms 
dX      df{ax  +  hy) 

J  dX      df(ax  +  ^y)      T  /v/  7  N 

It  is  easy  to  perceive  that  the  elimination  oi  f'{ax  +  hy) 

from  these  equations,  gives  the  equation  a-^ h-j-  =0,  the 

ay          cix 

same  as  in  1,  so  that  it  does  not  depend  on  the  form  of/*, 
lleciprocally,  if  in  any  calculation  we  meet  with  an  equa- 
tion of  the  form  a-j h-^  =  0,  it  may  clearly  be  supposed 

to  have  been  derived  from  an  equation  of  the  form 
X  =y (oa?  -f  &y)  =  an  arbitrary  function  of  oa?  +  Jy. 


DIFFERENTIATIONS   OF   XY.  27 

3.  If  a  =::  J,  the  preceding  equation  gives  --—=-—-;  con- 
sequently, if  X  —fi^x  +  y)  =  a  fanction  of  a?  +  y,  it  follows 
that  the  jMrtial  differential  coefficients^  when  taken  sepa- 
rately with  reference  to  x  and  y,  must  equal  each  other. 
For  more  ample  details  relatively  to  the  preceding  remarks, 
&c.,  we  shall  refer  to  Art.  77,  &;c.,  p.  230,  vol.  1,  of  the 
"Calcul  Differentiel  et  Integral,"  of  Lacroix. 

(11.)  If  X  and  Y  represent  functions  of  any  number  of 
independent  variables,  whether  the  variables  in  X  and  Y  are 
the  same  or  not ;  then,  we  propose  to  show  how  to  find  any 
differential  of  the  product  XY. 

Thus,  by  indicating  and  taking  the  differentials  of  XY  in 
succession,  we  get 

d  (XY)  ^  XdY  +  YdX, 

d?  (XY)  =  X^^^Y  +  2dX.dY  +  Yc?-X, 

c^^  (XY)  r-  Xd'Y  +  ZdX.d?Y  +  WXdY  +  c^XY,  ...  to 

d^ (XY)  :=  Xd'^Y  +  ndXd—'Y  +  -^%^^  cZ=X"--Y  + 

where  it  is  clear  that  n  denotes  an  integer,  such  that  the  nth 
differential  of  the  product,  indicated  by  (^"(XY)  in  the  first 
member  of  the  equation,  is  developed  in  the  second  member, 
and  of  course  the  equation  must  be  considered  as  being  an 
identical  equation. 

It  is  clear  that  {c)  can  be  obtained  immediately,  from  the 
development  of  {dY  +  dXy  according  to  the  descending 
powers  of  dY  and  the  ascending  powers  of  g?X,  by  the 
Binomial  Theorem  ;  being  particular,  in  the  development,  to 
apply  the  exponents  of  the  powers  of  o^Y  and  dX  to  the 


28  DIFFERENTIATIONS   OF  X,    Y,   Z,   ETC. 

characteristic  d,  and  to  write  Y  for  c?"Y,  and  X  for  d^X.  (See 
Art  91,  p.  256,  Vol.  1,  of  the  "Calcul  Differentiel,"  &c.,  of 
Lacroix.) 

Remarks. — 1.  It  is  clear  that  the  differentials  of  the  quo- 

X 

tient  ;r7u  =  XY~^  may  be  found  in  much  the  same  way  as 

before,  by  changing  Y  into  Y~\ 

2.  If  c?X,  dY^  dZ,  &c.,  stand  for  the  differentials  of  any 
number  of  functions,  X,  Y,  Z,  &c.  ;  then  the  differential  of 
the  product  indicated  by  c?"(XYZ,  &c.),  will  be  obtained  from 
the  power  {dX  +  dY  -f-  c?Z  -f-,  &c.)",  in  a  way  similar  to  that 
of  obtaining  the  differential  indicated  by  c?"(YX)  from 
{dY  4-  cZX)",  as  explained  above.  For  further  information 
on  what  has  been  done,  see  Lacroix  and  "Theorie  Analytique 
des  Probabilites  '^  of  Laplace. 

To  illustrate  what  has  been  done,  take  the  following 

examples. 

1.  To  find  the  differential  of  XY,  indicated  by  dXXY). 
Here,  by  putting  3  for  ?i  in  (c),  we  immediately  get 

c^^(XY)  =  Xd'Y  +  SdXd'Y  4-  ScPXdY  +  c^^XY : 

which  can  also  be  found  from 

(c^Y  +  dxy  =  {dYf  +  s{dYydX  +  sdY{dXf  +  (crxy, 

as  has  been  stated  ;  noticing  that  for  (dYf  =  1  x  {dYf,  we 
ought  to  write  [since  {dXf  =  1]  {clX)\dYf,  and  for  {dXf, 
we  must  also  write  {dXf{dYf. 

For,  by  changing  {dX)\dYf  into  Xd'Y,  and  S{dYfdX 
into  Sd-YdX,  and  so  on,  we  shall,  as  before,  get 

d'{XY)  =  Xc^^Y  +  SdXcPY  +  Sd'XdY  +  d'XY. 

2.  To  develop  tf (.?ry),  by  means  of  the  preceding  for- 
mula. 


DIFFERENTIATIONS   OF   PARTICULAR  EXAMPLES.         29 

Bj  tlie  substitution  of  ar*  for  X,  and  if  for  Y,  it  imme- 
diately changes  into 

d^ix'f)  =  \^;^^dxchf  +  Z^xyidxfdy  +  ^y\dx)\ 

since  x^d{dyY  =  0,  on  account  of  the  supposed  constancy  of 
dy,  y  being  regarded  as  an  independent  variable. 

3.  To  find  the  second  differential  of  -^-j  or  to  expand 
c?(XY-^),  when  X  =  a?^  and  Y  =  y. 

Here,  since  (^X^Y"^)  =  Xd\Y-')  +  2dXd{Y-^)  +  d'XY-\ 
by  putting  x^  for  X  and  y  for  Y,  and  performing  the  indi- 
cated differentiation,  we  get 

drix'y-^)  =.  Ix^irW  -  ^xy-Hxdy  +  ly-^dx"" 
Ijrdy'^         4:xd.xdy   .    2dx^ 

~~    f  f  y ' 

(12.)  If  ^  =  /'(s)  and  s  =  f'{x\  we  now  propose  to  show 
how  to  find  the  differential  coefficient  of  y  regarded  as  a 
function  of  x. 

Here  we  clearlv  have  dy  =    '  ,     dz,  and  dz  =  -~-  dx , 
^  d3        '  dx        ' 

and  consequently,  by  substituting  the  value  of  dz  from  the 
second  m   the   first,  we  have   dy  r=    '  -.      x  -^-r^    X  dx\ 

,  .  ,       .         ^y       dfiz)        dfix)  .     , 

which  gives  ~  =  '         x    -^^— ;   as  required. 

rt^/  CC3  CLX 

It  is  easy  to  perceive  that  if  y  =f{3),  z  =  (fi  (v),  v  =  i)  (a?), 
we  shall  in  like  manner  get 

"^^"^    dz    ""    dv     "^     dx    '^'^' 

Oj.  ^  ^  ¥(£)  ^  (Wl  ^  ^^H^) /^n  , 

dx         dz  do  dx     

and  so  on,  to  any  extent. 


80  EXAMPLES  (continued). 

EXAMPLES. 

1.  Given  y  =  3^^  z  =  4ji7',  to  find  the  differential  of  y  or 
its  differential  coefficient,  regarding  it  as  a  function  of  x, 

Here,  from  dy  =  Qzdz  and  dz  =  lIxHx^  we  get  by  sub- 
stitution, dy  =  72zardx,  or  -^  =  72^0^. 

2.  Given  y  =  az^,  z  —  W^  and  v  —  cx\  to  find  dy,  or  its 
differential  coefficient  regarded  as  a  function  of  x. 

Here,  we  have  dy  =  2azdz,  dz  =  Shv'dv,  and  dv  =  4:cx\lx ; 
consequently,  by  substitution,  as  in  (d),  we  sball  have 

dy  =  24^ahczv'Mx,  or  -f-  =  2'iahczv^a^. 
^  '        dx 

3.  To  simplify  the  differential  of  y  =  {aa?  -{-har  -\-  cf  or 
its  differential  coefficient,  by  putting  z  =  aa^  +  bx^  -j-  Cj  which 
reduces  the  proposed  equation  to  y  =  z\ 

Here,    from    y  =  z'^   we    have    dy  =  4:z\lZj     and    from 
z  =  aa^  -\-  ha?  +  G  we  have  dz  =  Saardx  +  2hxdx ;    conse- 
quently, from  the  substitution  of  dz,  we  have 
dy  =  4:z^  X  {Sax'^dx  +  2hxdx), 

or  ^^"^  ^^^'  +  ^^  +  of  X  {Sax'  +  2hx) ; 

which  is  the  same  result  that  the  immediate  differentiation. 
of  the  proposed  equation  will  give. 

Remarks. — 1.  Thus  we  perceive  how  we  may  often  sim- 
plify the  differentials  or  differential  coefficients  of  compli- 
cated expressions. 

2.  If  we  have  y  =f{ii.z),  sudh  that  we  have  u  =  (p(x) 

and  5  =  0  (a?) ;  then,  we  shall  clearly  have 

-        df(u.z)  ,         df(u,z)  , 

dy  =  -^ ^  du  +  -^S — ^  dz, 

^  du  ^        dz  ' 

d(l){x)  dxp{x)  y 

and  du  —     ,      dx,     dz  =      ,    ^  ax. 

itx  dx 


EXAMPLES  (continued).  31 

wHcli  are  clearly  the  same  as 

dv—-^  da  +  -f-  dz,   and  da  —  -,-  dx.    dz  = -j-  dx, 
^       da  dz      '  dx      '      '^       dx 

Hence,  eliminating  du  and  dz  from  cZy,  it  will  become 

dy      du   ^         dy     dz   , 

dy  _  dy     du       dy      dz 
dx  ~  da  '  dx       dz    '  dx' 

In  mucli  tlie  same  way,  if  we  have 

y  =  f(f,  V,  z,  &c.),  t  =  Y{x\  V  =  (p{xl  z  =  VH,  &c. ; 

then,  as  before,  we  shall  clearly  have 

dy_  ^di     dt_       dy     ^    .    ^    ^  +   &c.  .  .  .  (^). 
dx        dt   '  dx        dv  '  dx        dz'  dx      '       '  '  '  '  \  J' 

It  is  easy  to  perceive  that  we  may  use  (<?)  to  simplify  the 
differentiation  of  complicated  functions,  in  a  way  very  anal- 
ogous to  that  of  {d). 

Thus,  to  find  the  differential  or  differential  coefficients  of 

we  put  u  =  |/(a^  -f  x^),  z  =  \/(a^  —  a^),  and  thence  get 
y  z=  |/(w  —  z).     And 

da  —  dz        ,  xdx  ,    ,  xdx 

dy  =  K-Tr ^1    "^*  =— 77-T-^ — ^ :  ^^d-  dz  =  —  -— — 5-  ; 

^       2\/{a  —  zy  j^{a?-\-x^y  \/{a?—x')' 

consequently,  substituting  the  values  of  da  and  dz^  and 
restoring  the  values    of   u  and  z  in  \^{u  —  z\  we  have 

xdx  xdx 

-f- 


or 


32  DIFFERENTIATIONS  WHEN  THE 

the  same  that  the  immediate  differentiation  of  the  proposed 
equation  will  giva 

(13.)  Supposing  two  or  more  variables  to  be  connected  by 
any  equation,  or  that  each  of  the  variables  is  (in  virtue  of 
the  equation)  an  implicit  function  of  all  the  rest ;  then,  it  is 
proposed  to  show  how  to  find  the  differential  equation  that 
exists  among  the  variables ;  and  the  differential  coefficients 
that  result  from  considering  either  of  the  variables  as  being  a 
function  of  each  of  the  others. 

1.  Let  X  =  y^(a?,  y)  =  0,  stand  for  any  equation  between 
X  and  y ;  then,  since  X  may  clearly  be  treated  as  being  an 
explicit  function  of  x  and  y  considered  as  being  independent 
variables,  we  shall  have 

.^^       c?X   -        c?X   , 

or,  since  dX  =  0,  the  equation  is  reduced  to 

dX    .        i 
^^  +  - 

From  this  equation  we  have 


dX    .        dX.         _ 


dx  __        dy 

dx 

in  which  x  is  regarded  as  a  function  of  y :  and,  by  taking  the 
reciprocal  of  this  equation,  we  have 

dy  dx  '        ( j!\ 

dy 
in  which  y  is  regarded  as  being  a  function  of  x. 


FUNCTIONS   ARE   IMPLICIT.  33 

Tlius,  if  X  =  y  —  3a?  we  have 

—   =   ^  ^  1  and   —    =  -  3 
dy  dij  ~    ^^        dx  ' 

wMcli  reduce    the    first    of   the    preceding    equations    to 
-^  =  -  and  the  second  to  -j-  —  3.     It  is  easy  to  perceive 
that  the  equation  y  —  3a?  =  0  or  2/  =  3a?,  immediately  gives 
<^a?  _  1       ^y  _  Q 

the  same  as  before,   and  found  in   a  much   more   simple 
manner. 

Similarly,  from  X  =  a?^  -f  y^  —  r'  =  0  we  get 

consequently,  we  have 

dx  __        dy  _       y  dy  _        x 

dy  ~~       dX  ~       X  dx  ~        y' 

dx 


Again,  by  differentiating  the  equation  x^  -{-  y"^  ~  r^  =  0 

)  have    z 
dy  _       X 


dx  u 

we  have    2xdx  +  2ydy  =  0 ;    which  gives   -j-  =  —  -     or 


-    _  ,  the  same  as  before. 

dx  y 

2.  Supposing  X  =  0  to  be  a  function  of  any  number  of 
variables,  a?,  y,  s,  &c.,  then  (as  before)  we  shall  have  the 
differential  equation 

-y-  dx  -\-   -^  dy  +   —7—  dz  +,  &c.,  =  0. 
<2a?  dy     ^         dz  '       ' 

Supposing  z  to  be  regarded  as  being  a  function  of  each 

of  the  other  variables,  then  the  partial  differential  equation 
2* 


dX. 

dz 
dx  ~ 

dx 
dz 

34  ELIMINATION  OF  CONSTANTa 

between  x  and  2-,  gives  -^  dx  +   -—  dz  —  0, 


which  gives 


dX. 

and  in  like  manner,  -7^  = -,^  ,  and  so  on. 

dy  dX 

Kemark. — T/ie  elimination  of  a  constant  from  an  equa- 
tion hy  means  of  its  differential  equation^  generally  changes 
the  form  of  the  differential  coefficient. 

Thus,  by  taking  the  differential  of  y'  =  ax^  we  get 

*lydy  =  adxj 

which  gives  a  =    '^     ; 

consequently,  substituting  this  value  for  a  in  the  proposed 
equation,  we  have 

2.d^  =  ydx      or       J    =^. 

Hence  the  differential  coefficients 

dy         a  T     dy         ?/ 

-f-  =  ~    and     -JL  —  J_ 
dx        2y  dx        2x 

although  equivalent,  are  of  different  forms. 

(14.)    When  a  variahle  is  a  function  of  any  variables 

regarded  as  independent 'y  then^  in  taking  tlie  second^  third^ 

(&c.,  differentials  of  the  function^  the  differentials    of  its 

independent  variables  must  each  be  constant  or  invariable. 

What  is  here  said,  is  clear  from  what  is  shown  at  pages 


V 


DIFFERENTIALS   OF  THE   SAIME  DEGREE.  35 

11   and  12,   mid  clearly  results  from  the  nature  of  the 
case. 
Hence,  we  deduce  tlie  following  conclusions : 

1.  The  ntli  differential  of  an  explicit  function  of  any 

number  of  independent  variables,  is  either  equal  to  the  sum 

of  terms,  that  contain  n  dimensions  of  the  differentials  of 

the  independent  variables,  or  it  is  equal  to  naught. 

a? 
Thus,  if  y  =  -,  and  x  is  the  independent  variable,   we 

have  dy  =  ?^,  d'y  =  ?1^,  d'y  =  0,d'y  =  0;  all  thedif- 

ferentials  above  the  second  being  equal  to  naught,  since  x 
Deing  the  independent  variable  dx  is  constant  or  invariable. 

2.  T/ie  nth.  differential  of  an  implicit  function  of  any 
number  of  independent  variables  mixed  together,  must  be 
such  that  there  shall  be  n  differentials  in  each  of  its  terms. 

Thus,  from  yx  =  c^,  regarding  y  and  x  as  functions  of 

other  variables  ;  we  have 

ydx  -f  xdy  =  0, 

xd^y  +  2dxdy  +  yd'^x  =  0, 

xd^y  +  Mxd^y  +  Sd'xdy  +  d^xy  =  0, 

and  so  on,  as  at  page  27. 

If  we  proceed  in  like  manner  with  the  equation 

y'^  +  a^  —  r^  =  0, 

we  get  2yc?y  +  2xde  =  0 

(for  which  we  may  put  ydy  +  xdx  =  0), 

yc^y  +  dy-  -f  xd^x  +  dx^  =  0, 

yd^y  +  Sdyd-y  +  Sdxd'x  +  xd!^x  =  0, 
and  so  on. 

If  »  is  taken  for  the  independent  variable,  or  if  dx  is  con- 
stant or  invariable,  these  equations  will  be  reduced  to  the 


SQ  CHANGING  THE  INDEPENDENT  VARIABLE. 

more  simple  forms  ydt/  -f  xdx  =  0, 

ydy  +  d7/  +  cZ^  =  0, 

and  yd^i/  H-  Sdydry  =  0. 

In  like  manner  if  x  is  regarded  as  a  function  of  y,  we 
have  ydy  4-  wdx  ==  0, 

xd^x  +  dx^'+dy^  =  0, 

a7cf a?  +  3c?a?cZ"a?  =  0, 
and  so  on. 

Again,  if  x  and  y  in  tlie  product  xy^,  are  considered  and 

treated  as  being  independent  variables,'  then  we  shall  have 

d  {xy^)  —  yHx  +  2xydy^ 

d\xy')  =  4:ydydx  -f  2xdif 

d^^X'tf)  =z  Qdxdi/j 

d\xf)  =  0, 

d%X2/)  =  0, 

and  so  on. 

(15.)  Having  an  equation  between  x  andy^  in  which  x  is 
the  independent  variable  /  we  propose  to  show  how  to  change 
the  equation^  so  that  x  sJiall  hecome  a  function  of  y,  or  so 
that  X  and  y  shall  hecome  f  motions  of  a  new  variable. 

We  may,  according  to  what  is  shown  at  pages  10  and  12, 

represent  that  y  is  a  function  of  x  by  the  form  -—-  =  -7--  ; 

dy 
consequently,  we  may  differentiate  the  first  member  of  this 
by  regarding  x  as  being  the  independent  variable,  and  the 
second  member  on  the  supposition  that  a?  is  a  function  of  y ; 

which  gives  -7^  = ■^-~ .     It  is  easy  to  perceive  that 

this  result  is  the  same  as  to  write  d  ~  for  -^  ,  and  then 

ax  dx 


CHANGING  THE   INDEPENDENT  VARIABLE.  37 

to  differentiate  d  -~  hj  regarding  dy  as  constant,  or  taking 

y  for  tlie  independent  variable. 

Indeed,  if  we  had  differentiated  tlie  right  number  of  the 
above  equation  by  regarding  dx  and  dy  as  both  variable,  we 

should  have  found  c?    —    =    ^^^  ^  -,  J^    ^j  which  is  clearly 

\dx\-  dor  '  "^ 

dij 
the  same  as  to  take  the  differential  of  —— ,  when  dy  and  dx 

CiX 

are  both  regarded  as  variable;  consequently  if  we  make 
d"y  —  0,  or  dy  —  const.,  the  preceding  differential  reduces  to 
d'^xdy 


d'j? 


,  as  found  above. 


It  is  hence  manifest  that  for  -—  we  may  write  d  -~  ^  or 

regard  dy  and  dx  as  both  variable,  or  if  convenient  take 

d-xdy         T     . 

—-,-  :  and  vice  versa. 

dx^ 

(16.)  To  show  the  facility  that  the  use  of  partial  differen- 
tial coefficieoits  sometimes  gives  in  the  solution  of  difficult 
questions  J  we  will  take  the  following  important 

PROBLEM. 

Given  0  =  0  (a,  +  xy)  —  a  function  oi  a  -{-  xy  (1),  in 
which  a  and  x  are  regarded  as  independent  variables,  and 
y  —  i^  (2;)  =  a  function  of  z  (2) ;  then  it  is  proposed  to 
develop  2  according  to  the  ascending  powers  of  x. 

According  to  Maclaurin's  Theorem,  see  (h)  page  17,  we 
may  assume 

,       dz'  d?z'    ^        d^z'    X? 

,"^"  +  5^  ^  +  ^T2  +  ^  12:3 +'^^-  •  •  (^)' 


38  THEOREMS  OF 

for  the  development,   in  which  z\    ^— ,   -7^- ,   &c.,   repre- 

dz    iP'z 
sent  the  values   of  s,   -r- ,  -73  ,  &c.,  that  result  from  put- 

OjX     CLUCi 

ting  a?  =  0  in  them. 

It  is  manifest  by  taking  the  partial  differential  coefficients 
of  (\\  that  we  shall  have 

dz  _  d.<p{a  -f  xy)       d.{a  +xy) 
dx  ~    d.{a  +  xy)  dx       ' 

,  dz  _  d.(f){a  4-  xy)        d.{a-{-  xy)  ^ 

da         d.{a  -f-  xy)  da        ' 

consequently,  eliminating  the  difierential  coefficient 
o?.<^(a  +  xy) 
d.{a  +  xy) 
from  these,  we  shall  have 

dz       d.{a  4-  xy)  _  dz       d.{a  +  osy) 
dx  da         ~  da  dx        ^ 

dz/  da\       dz  /  dy\ 

dx\  dyl       da  \  dx)  ' 

Because  (2)  gives 

dy  __  d^{z)  dz  _dy     dz  ,     dy  _  dy    dz 

da~     dz     da~~  dz  '  da^  dx^  dz  '  dx^ 

we  easily  reduce  the  preceding  equation  to 

dz         dz  ,.. 

di=yd^ (^> 

By  taking  the  partial  differential  coefficients  of  (4)  rela- 
tively to  a?,  we  shall  have 

d'^z        -,(     dz\        ,        dy    dz   ,       ,  (dz\        , 
d^  =  '^[yd-a)-^^  =  tx'd^^^^\d-a)--'^^'^ 
or,  since 

dy      dy     dz  ,         ,  /dz\        ,  drz  drz 

-f  —~f  .  -—    and    ydl-y-)  -T-dx  =  y~J—^=y-^-^ 
dx      dz     dx  ^    \dal  ^  dadx     ^  dxda 


LAPLACE  AND  LAGRANGE. 


39 


(page  22),  we  have 

cPz       dy    dz    dz  d/z 

r=   —  -  .  . \-  y 

ddi?       dz    da    dx     •    dxda 


(dy  dz       dy\    ,/     dz\        , 


'^'='^(2''£)-^'^'^  = 


dz  dz 

and  since  (4)  gives  -j-  ^=  y  -j--,  this  is  easily  reduced  to 

dx  CLCL 

'^ 

dx^      ^V  daJ    '    ^^  da 

Differentiating  the  members  of  this  equation  relatively  to 
iB,  we  have 

^.  -(43  -¥'£}  ,M4J 


dadx 


dxda 


dx 


-7-  day 


on  account  of  the  independence  of  a  and  a?,  and  the  differen- 
tiations relatively  to  them. 

It  is  easy  to  perceive  that may,  as  before,  be  re- 
duced toe?  1 2/^-^1 -f- o?a,  which  gives 

and  proceeding  with  this,  as  before,  we  have 


d^_ 

dx'~ 

which,  as  before,  gives 


(^2 


Pi): 


dx 


da"", 


S^^K^'S^^"''^''^'''^'^ 


40  THEOREMS  OP 

If  the  values  of -i—  and  y,  that  result  from  putting  a?  =  0 
in  them,  are  represented  by  -^  and  y\  we  shall  have 

dz'        ,dz'    d^-z'      ^y'db)        , 
Hence,  from  the  substitution  of  these  values  in  (3),  we  get 

"="+ 1^' rf^)  *+ -^^- 1:2  + —d^- 1:2:3 +•  *^ 

■ (A); 

which  clearly  holds  good,  when  any  like  functions  of  z  and  z' 
are  put  for  z  and  z'  in  it ;  noticing,  that  (A),  thus  generalized, 
is  called  the  Theorem  of  Laplace  /  and  if  we  put  1  for  a?,  in 
(A),  it  will  become  what  is  called  the  Theorem  of  La  Grange. 
To  perceive  some  of  the  uses  of  (/i),  take  the  following 


EXAMPLES. 

1.  Given  'bz'^  —  cz  -\-  d^=^  0,  to  find  2  in  a  series  of  the 
known  quantities. 

The  equation  is  readily  changed  to  the  form 

consequently,  for  0  in  (1)  we  put  1,  or  unity,  and  y  =  s" ; 
also,  a  =   -     and    x  =  -. 

By  putting  a?  =  0  we  get  z'  =^  a  and  thence  -^  =  1 ; 
also,  y  =  z'""     gives    y'  =  s'"  =a", 

and  thence  y  -5—  =  a". 


LAPLACE  AND  LAGRANGE.  41 

Because  y  =  a"     and      ^  =  1' ^  (2/'' ^) -^  ^* 
becomes  \     ■  =  2;2a2"-' 

also,  ^^  (2/^^ ^)  -r-  ^ct^  =  6^'  (a^'O  ^  ^a^  =  8/1  (3/i  -  1)  a^"-^ 
and  so  on. 

Hence,  collecting  the  results,  we  shall  get 

z  =  a-\-a^x  +  2na^'^-^    ~  +  3?i(3?i  -  l)a^"-^  j^  +,  &c. 

If  7^  =  3,  5  =  1,  c  rrr  3,    and    d—  —  1, 

tlie  proposed  equation  becomes  ^'^  —  3^  —  1  =  0 ; 

whicli  gives  a  =  —  ^ ,  and  a?  =  -^  . 

Hence,  from  tbe  preceding  series,  we  shall  have 

_  _  1  _  jL 1 £_  _ 

^~      3      81        729       19683      ' 

-       ^^^^     ,&c.  =  - 0.3172, 


19683 

which  is  one  of  the  roots  of  the  equation  s'^  —  3^  —  1  =  0 ; 
correctly  found  in  all  its  figures. 

2.  Given hz—  cz""  -\-  d—  0,  to  develop  s in  a  series. 
Since  the  equation  is  equivalent  to 
d      <?     1 

we  have  ^  —  \^a=  — y-,a?  =  -7,y  =z  n  .    Putting  a?  =  0, 

dz'  i  i        1 

get  z'  =a  and  -^  =  1 ;   and  y  =:  z^^  becomes y^  =z'  »  =a". 


42  THEOREMS  OF 

Hence,  if  we  change  n  into  -  in  the  series  for  s  in  the  pre- 
ceding example,  we  shall  get 

n  1.2       n\n         J  1.2.3  ' 

as  required. 

If  we  have  the  equation  3»'  —  -y  —  1  =  0 ;  then,  putting 

v^z^z    or    -y  =  2^ 

it  becomes  3^  —  s*  —  1  =  0    or    ^  =  o  +  b  ^  > 

o        o 

SO  that  n  =  3,   a  =  77    and    a?  =  ^  in  this  equation. 
If  -  is  put  for  -  a  and  x  in  the  series  for  ^,  it  becomes 


^-e'-ey-."'^ 


=  0.3333  +  0.23112  +  0.053416  -f,  ifec.  =3  0.61787  +,  &c. ; 
and  hence  v  =  I/3  =  1^0.61787  =  0.85173,  whose  first  two 
decimal  places  are  correct 

3.  Given  As"+  'Bz'"+  Cz^^-h  . . . .  +  N  =  0,  to  find  2. 

Since  the  equation  is  equivalent  to 

2*^  =  -  ^  -  -^-  (B3"'  +  C2^''  +,  &c.)  =  a-\-xy; 

N  1 

we  have  a  =  — v-  and  i»y  = r-  (B3"'  +  C3"''4-,  &c.);  and 

-A.  -A. 

we  may  evidently  put  x  = r-  and  y  =  Bz^'-^-  03"''  +  ,  &c. 

From  what  precedes,  we  get  z  =  {a  +  xi/Y,  which  corre- 
sponds to  <^  (a  +  xi/)  in  (1),  p.  37  ;  which,  by  putting  x  =  0, 

gives  z'  =  a^^j  which  gives 


LAPLACE 

\  AND 

LAGEANGE. 

I  — M 

chJ 

1     1_ 

-  a« 
n 

da 

n 

43 


and     y'  =  B^''^'  +  Qz"'"+,  kc.  =  Ba'^  +  Ca"  +,  &c. 
Hence,  from  (h)  we  get 

l-n 
1  £'  r»^'  ^^    n 

s  =  a"  +  (Ba"  +  Ca''*"  +,  «&c.)  -—  x  + 

nf  n"  \  —  n 

d  [(Ba"^  +  Ca»  +,  &c.y  a"^~]     x" 

da  1.2n^       ' 

v'  n"  \  —  n 

cZ-^[(Ba"^+Ca"+,&c.ya  "]      ^        ,     ^ 
~'  da?  1.2.3  7.  "^'  ^''• 

To  illustrate  ttis   formula,  we  shall  take  the   equation 

^3  _  3^  _  1  ^  0,  under  the  form  z"  —  z-^  —  3  =  0. 

Hence,        A  =  1,   B  = -1,   C  =:  0,   D  =  0,    &c., 

N  =  -3,  a^-^^Z,x=.-\  n=2,  n'^  -1,  n"  =  0,  «&c. 
From  the  formula,  we  get 

2?iV.   "" 

consequently,  by  putting  3  for  a  and  2  for  n,  and  giving  the 
square  roots  the  ambiguous  sign  ± ,  we  get 

2  =  ±  V3  +  g±  ^  +,&c. 

=:  ±  1.7320  +  0.1666  T  0.0138  +,  &c. 
Hence,  we  have   1.88  +    and    —1.54—   for  approximate 
values  of  two  of  the  roots  of  the  proposed  equation,  cor- 
rectly found  to  two  places  of  figures  in  each. 

Eemarks. — 1.    It  is  sometimes  necessary  to  distingiiisli 


44  DIFFERENT  METHODS. 

between  total  and  partial  differential  coefficients.     Thus,  if 

-  du du  dp      du  dq       du  dr 

dx  ^  dp  dx      dq   dx       dr    dx ' 

we  call  the  first  member  of  the  equation  the  total  differential 
coeffeieiit^  and  the  terms  that  compose  its  right  member  are 
its  parts,  or  what  are  called  the  partial  differential  coeffi- 
cients. 

2.   li  p  =  a?,  it  is  clear  that  the  equation  will  be  reduced 

du  _  du        du  dq        du    dr 
dx        dx        dq  dx        dr  dx'' 

where  it  will  be  perceived  that  the  total  coefficient  -7-  in  the 

first  member  of  the  equation,  is  apparently  the  same  as  the 
partial  quotient  in  the  second  member;  consequently,  for 
distinction's    sake,   we    inclose   the    partial   quotient  in   a 

parenthesis,  thus  (-j-)-  Hence,  the  preceding  equation  will 
be  written  in  the  form, 

du  _  ldu\       du  dq         du  dr  ^ 
dx  ~~  \dxl       dq  dx         dr  dx  ' 

and  we  may  clearly  proceed  in  like  manner  in  all  analogous 
cases. 

(17.)  It  may  not  be  improper,  in  concluding  this  section, 
to  notice  some  of  the  different  methods  that  have  been  used 
by  different  authors  in  treating  the  Differential  Calculus. 

1.  Leibnitz  and  Newton,  the  illustrious  founders  of  the 
Calcalus  under  different  forms,  respectively  used  the  infin- 
itesimal method^  and  that  of  the  liiniting  ratio. 

Thus,  to  find  the  differential  of  x^ ;  we  change  x  into  x-\-h 
and  thence  get  {x  -\-  hf  —  ar^  =  Sx^h  +  Sxh^  +  A^,  for  what  is 
generally  called  the  difference  of  x^ ;  noticing,  that  it  is  some- 


DIFFERENT  METHODS.  45 

times  called  tlie  increment  or  decrement  of  x"^  accordingly  as 
it  is  positive  or  negative. 

If  K  is  finite,  the  difference  being  evidently  finite,  is  called 
2^  finite  difference  /  and  is  often  denoted  by  writing  the  Greek 
letter  J  (delta),  called  the  cJiaracteristio  of  finite  differences^ 
before  or  to  the  left  of  x^ ;  and  since  h^=x  +  h  —  x^  we  write 
Ax  for  h  ;  consequently,  for  {x  +  hj—  x^=  dx'/i  +  Sxh?  +  h% 
we  may  write  Jx'^  =  Sx"Jx  +  3x{Ax)-  +  {Jxf :  noticing,  that 
x^  or  (more  generally)  x"^  -^  c,  c  being  constant,  is  often  called 
the  integral  of  Jx^  or  of  its  equivalent,  Sx^Ax  -\-  SxJj^  +  Ax\ 

If  h  is  unlimitedly  small,  or  an  infinitesimal,  it  is  clear 
that  Bx^h  +  Sxh^  +  h^  will  also  be  unlimitedly  small,  or  an 
infinitesimal ;  and  if  infinitesimal  differences,  sometimes 
called  dvfferentials^  are  distinguished  from  finite  differences 
by  writing  d  for  J,  tJien^  according  to  the  method  of 
Leibnitz,  the  equation  {x  +  hy  —  ar'  =  3a?-A  +  Bxh^  +  h^  be- 
comes dd(^  =  Sardx  +  Sxdx^  +  dx^ ;  for  which,  on  account  of 
the  comparative  minuteness  of  Zxdx^  and  da^^  toe  may  evi- 
dently write  dd^  =  Sx^dx,  which  is  of  the  same  form,  that 
our  rule  at  p.  5  will  give  for  the  differential  of  a^ :  noticing 
that  a?^  +  c  is  called  the  general  integral  of  dx\  or  of  its 
equivalent,  Sx^dx. 

To  signify  that  the  integral  of  any  finite  difference  is  to 
be  taken,  the  Greek  letter  ^  (sigma)  is  generally  written  be- 
fore or  to  the  left  of  the  difference,  inclosed  in  a  parenthesis, 
if  necessary.  Thus,  ^Jx"  =  I  \Zx\Ax)  +  Zx^dxJ  +  {AxJ'], 
which  clearly  equals  SIx^Jos  -f  S2:x{Axy  +  ^Axf^  is  used  to 
denote  that  the  integral  of  Ax^,  or  of  its  equivalent, 

3x%Ax)  +  dx{Jxf  +  {Axf  , 
is  to  be  taken  ;  and  since 

(x  +  h)^  —  x"^  —  {iv  4-  hf  -\-  G  —  x^  —  c=  A{x^  +  c\  c  —  const: 
the  most  general  form  of  the  indicated  integral  is  a?^  +  a 


46  DIFFERENT   METHODS. 

In  much  the  same  way  we  indicate  the  integral  of  any 
proposed  differential,  by  writing /,  called  the  sign  of  integra- 
tion, or  the  characteristic  of  integrals^  to  the  left  or  before 
the  diflereflllial,  as  before.     Thus,  we  have 

in  which  c  =  coast,  =:f'dx-d:c  =  Zfj^ddn  =  a?'  +  c:  noticing, 
that  the  constant  c  is  used  for  generality,  or  to  make  the 
integral  applicable  to  any  case  tliat  may  be  required. 

Again,  resuming  {x  -f  hf  —  x^  =  3xVi  +  Sxh^^  +  A^,  and 
dividing  its  members  by  A,  it  will  become 

/I 

which  clearly  shows  if  h  is  diminished  indefinitely,  the  right 

member  has  Sx-  for  its  limit 

Hence,  according  to  the  common  method  of  taking  the 

limit,  by  putting  h  =  0,  the  equation  is  reduced  to  the  form 

0  .0  dx^ 

-  z=Sx'\  or  since  for  -  we  ought  evidently  to  write  ~  we 

d^ 
have -7-  =  Sx'dx ;  see  my  Algebra,  pages  256  and  257. 

Since  {Sx"" h  +  ^x/i^  +A-^)  -^h=Sa^  -\-Zxli'  +A^  tliis  quotient 
is  often  (with  great  impropriety)  called  the  ratio  of  the  incre- 
ment or  decrement  of  y?  to  the  corresponding  increment  or 
decrement  of  the  independent  variable  x ;  and  3.2?^,  the  limit 
of  the  quotient,  is  often  improperly  called  the  limit  of  the 
ratio  when  h  is  infinitesimal. 

The  preceding  process  in  substantially  the  same  as  Newton's 
method  of  limits. 

Because  [a  (x  +  A)"  +  c  —  {ax''  +  c)] 

=  a\nx''-'h+  ^^^^— ^ X  "-» h'  +  ,  &c.] 

1.  Ji 


DIFFEKENT   METHODS.  47 

is  Tinder  the  form  of  an  exact  difference,  if  h  is  finite,  the 
equation  (agreeably  to  what  has  been  done)  can  be  expressed 
by  the  form 

A{(ix^  +  c)  =  a  \nx''-^Ax  +  ^^^V^— ^"~'  (^•^)'  +  »  ^^.J  ; 

1 .  Z 

hnt  if  h  is  infinitesimal,  the  equation  is  equivalent  to 

diax"^  4-  c)  =  nax'^~'^dx. 

Similarly,  because  {x  -f  h)  (]/  +  k)  —xy^=  xh  +  yh  +  Kh  is 
under  the  form  of  an  exact  diJfference,  if  h  and  k  are  finite, 
the  equation  may  be  expressed  by  the  form 

A  {xy  -\-  c)  —  xAy-^yAx  +  AxAy; 

but  if  h  and  k  are  infinitesimals,  the  equation  becomes 

d  (xy  +  6')  =  xdy  +  ydx ; 

by  rejecting  dxdy  on  account  of  its  comparative  minuteness. 

It  is  manifest  from  these  examples,  that  in  order  to  find 
the  integral  of  any  finite  difierence  or  differential,  it  must  be 
exact,  or  be  reducible  to  a  difference  or  differential  which  is 
either  exact,  or  differs  insensibly  from  an  exact  difference  or 
differential. 

2.  Eesuming  the  equation 

{x  +  hf  -0^  =  ^xVi  +  Sx/r  +  h^ , 

and  putting  dx  for  k  in  the  first  term  SjcP/i  of  the  difference 
of  ./,  it  will  become  Sx^dx.  If  the  operation  to  be  per- 
formed on  x^,  in  order  to  obtain  Sx'^dx  from  it,  is  denoted  by 
d.  x^  or  dx^,  we  shall  have  d.x^  =  dx^  —  3x'dx ;  which  indi- 
cates and  expresses  the  differential  of  ar',  obtained  by  defi- 
nition, accordlnfj  to  the  method  proposed  hy  the  celebrated 
Lagrange. 

Supposing  X  to  be  any  function  of  a?,  and  that  X  becomes 
X'  when  x  is  changed  into  x  ±,h\  then,  supposing  x  and  h 


48  DIFFERENT  METHODS. 

to  be  undetermined,  Lagrange  proved  that  X'  may  be  ex- 

pressed  by  tbe  form  X  ±  X,  A  +  Xo  7-^  ±  X3  — —  +  ,  &c.; 

in  which,  he  called  Xi,  X2,  X3,  &c.,  the  first,  second,  third, 
&a,  derived  functions  of  X ;  and  it  is  easy  to  perceive  that 
the  series  is  the  same  as  Taylor's  Theorem. 

3.  The  difficulties  and  unsatisfactoriness  that  have  attended 
the  treatment  of  the  first  principles  of  the  Differential  Calcu- 
lus, appear  to  us  to  have  arisen  from  the  circumstance,  that 
it  has  been  thought  necessary  to  convert  X'  into  a  series  of 
the  form  X  +  Ah  +  A^A'^  -f-  AJi^  -f  ,  &c.,  and  then  to  reduce 
the  difference  X'  —X  =  A^  +  A/r  +  A.A^  +  ,  &c.,  to  its  first 
term  AA,  in  order  to  get  dX.  =  Kdx ,  or  the  differential  of  X. 
For  this  process  has  evidently  introduced  the  infinitesimals 
of  Leibnitz,  and  the  limiting  ratios  of  Newton  and  others, 
into  the  Calculus,  as  furnishing  reasons  why  the  terms 
A/i-,  AgA^,  &c.,  must  be  rejected,  in  comparison  to  A  A. 
Whereas,  the  true  reason  for  the  omission  of  these  terms,  is 
that  so  long  as  x  and  h  are  indetermi  nates,  the  term  A  A  rep- 
resents the  sum  of  all  the  changes  of  X  that  result  from  the 
separate  change  x'  —  oj  =  A  of  each  x  contained  in  X . 

And  it  is  evident  from  the  reasoning  in  (9)  at  p.  15,  that 

we  may  consider  the  terms  that  follow  the  second  term  -7-^  h 

in  Taylor's  Theorem,  as  deducible  from  it  when  x  and  h  are 
regarded  as  being  inde terminates,  in  a  way  very  analogous  to 
that  of  finding  the  terms  that  follow  the  second  term  from  it, 
in  the  investigation  of  the  Binomial  Theorem:  see  Ex.  16, 
p.  56,  of  my  Algebra. 


SECTION  IL 


TRANSCENDENTAL   FUNCTIONS. 

(1.)  "When  a  function  is  sncli  tliat  it  can  not  be  expressed 
by  means  of  its  variable  and  constants  in  a  finite  number  of 
algebraic  terms,  it  is  called  a  transcendental  function.  Thus, 
log  a?,  a^,  sin  a?,  cos  a?,  &c.,  are  transcendental  functions :  the 
first  being  a  hgaritJmiic  function,  tbe  second  an  exponential 
function,  and  the  third  and  fourth  are  circular  functions. 

(2.)  Any  number  or  quantity  may  he  expressed  in  a 
transcendental  form. 

For  if  a  represents  any  number  or  quantity,  it  is  clear  that 
for  a  we  may  write 
/         a^      a""      a^        ,     \       I         a"-      a^      a^        o    \^  1 

/  a"-       a^       a^         c     \'      1 

+  r-T+T-T+'M  1:2:3+'^'- 

.       ,  .  ,  a^        a'        a' 

m  which  ^~T"*""3 X+'^^-' 

is  called  the  hyperbolic  or  Napierian  logarithm  of  1  +  a. 

c^       a^        a^ 
Hence,  if  we  put  a 9-  +  "o"  — j-  +,  &c.  =  A,  we  shall 

A"        A^ 
clearly  have  l4-a  =  l  +  A  +  — -  +  — --—  +,  &c. ;  and  in 

52  l^  ^4 

like  manner,  if  5  —  —  +  — —  +,  &c.,  is  represented  by 

B,  we  shall  have  l  +  5=::l  +  B  +  ^+  -^-^l  +'  ^^ 
3 


50  TRANSCENDENTAL  FUNCTIONS. 

(3.)  The  product  of  the  correspondinff  members  of  these 
equations  will  be  of  a  similar  form. 
For  we  shall  clearly  have 

(1  +  a){l  +  h)  =  l  +  a  +  h  +  ah  =  l  +  (A  +  B)+  (A  +  B)^  j^ 
+  (A  +  Byji^+,&c. 

If    a-i-h  +  ah  —  ^ ^ +   3 ^— ,  &c., 

is  represented  by  C,  it  is  clear  from  what  has  been  done,  that 
the  preceding  equation  is  equivalent  to 

^  +  ^  +  r2  +  ui+'^ 

=  H-A  +  B  +  (A  +  B)=j?2+(A  +  B)»jl3+,&a; 

which  clearly  gives  C  =  A  -f-  B. 

Because  A  and  B  are  the  hyperbolic  logarithms  of  1  +  a 
and  1  +  h,  and  that  C  is  the  hyperbolic  logarithm  of  their 
product,  it  results  from  the  preceding  equation,  that  the 
hyperbolic  logarithm  of  a  product  equals  the  sum  of  the 
logarithms  of  its  factors.  • 

If  the  members  of  C  =  A  +  B  are  multiplied  by  the 
arbitrary  multiplier  m^  called  the  rnodulus ;  it  is  clear  that 
its  properties  will  not  be  changed,  and  we  shall  get 

mC  =  mA  +  77?B; 
such,  that  mA,  w-B,  and  wiC  may  be  called  logarithms  of 
1  +  a,  1  +  ^,  and  of  their  product. 

Hence,  in  any  system  of  logarithms,  the  logarithm  of  a 
product  equals  the  sum  of  the  logarithms  of  its  factors  / 
reciprocally,  tfte  logarithm  of  a  quotient  equals  the  logarithmt 
of  the  dividiind,  minus  that  of  the  divisor. 

Hence,  too,  tJie  logarithm  of  a  power  equals  the  logarithm 
of  its  root  multiplied  hy  the  index  of  the  power ;  and  re- 


LOGARITHMIC  FORMULJB.  51 

ciprocallj,  the  logarithm  of  a  jpower^  divided  hy  its  index^ 
equcds  the  logariihm  of  its  root. 

If  the  logarithm  of  a  number  or  quantity,  whose  modulus 
is  m,  is  indicated  by  writing  log  before  or  to  the  left  of  it 
(inclosed  in  a  parenthesis  when  necessary),  we  shall  clearly 

have  log  (1  +  a)  =  m  /a  —  ^  +  -|-  —  ^  +  ?  &c.  j (a) ; 

which  we  shall  call  the  Logarithmic  Theorem. 

It  is  evident  from  what  has  been  done,  that  we  shall  have 

(l  +  a)'  =  l  +  A,«+(^J^+^^J+,&c (J); 

which  is  called  the  Eicponential  Theorem^  in  which  A  and 
Kx  are  the  hyperbolic  logarithms  of  1  +  a  and  (1  +  a)^. 
If  A  =  1,  Q>)  becomes 

.*....  (^0; 

which  gives      . 

l  +  «=l+l  +  ^  +  j^  +,&c.  =2.7182318284  +,&c. 

which  is  generally  expressed  by  ^,  and  is  called  the  hase  of 
hyperholic  logarithms^  since  its  hyperbolic  logarithm  is  sup- 
posed to  be  unity  or  1 ;  consequently,  putting  6  for  1  +  «  in 
{l)'\  it  becomes 

,«  =  !  +  , +  i5+j|_+,&o. r): 

which  shows,  if  we  put  e^  =  N,  that  we  shall  have  x  =  the 
hyperbolic  logarithm  of  N,  since  that  of  ^  =  1. 

If  we  write  log  before  a  number  or  quantity  (inclosed  in  a 
parenthesis  if  necessary)  to  denote  its  hyperbolic  logarithm, 
it  is  clear  that  log  (1  +  d)^  =  Ax ;  and  as 


62  LOGARITHMIC  FORMULA. 

log  (1  4-  ay  =  7n  log  (1  +  af^ 
we  get  log  (1  +  ^y  =  rnAx. 

If  we  assume  m  A  =  1^  or  m  =  -r-,  the  preceding  equation 

becomes  log  (1  +  ay  =  x,  and  of  course  log  (1  +  a)  =  1 ; 
consequently,  1  -f  a  represents  the  base  of  the  logarithms 
denoted  by  log.  Hence,  assuming  (1  +  ay  =  N,  we  have 
log  (1  +  ay  =  log  N  =  a? ;  since  1  +  a  is  supposed  to  be  taken 
for  the  base  of  the  logarithms  represented  by  log.- 

Because  log  N  —  m  log  N  =  — ^ — ,  it  results  that  vje 

shall  get  log  N,  hy  dividing  the  hyperholic  logarithm  of  N 
hy  the  hyperbolic  logarithm  of  the  hase^  or  hy  inAiltijplying  it 

hy  the  modulus  (-r)'-,  reciprocally,  log  N,  multiplied  hy  the 

hyperholic  logarithm  of  the  hase^  or  divided  hy  the  modulus^ 
gives  the  hyperholic  logarithm  of  N". 

Thus,  if  the  base  1  +  a  =  10  =  the  base  of  common 
logarithms,  the  tables  of  hyperbolic  logarithms  give 
log  10  =  2.3025850929,  and  thence  the  modulus 

(m)  =  \  =  ^—  =0.4342944819. 
^    ^       A      log  10 

Again,  from  the  tables  we  have  log  2  =  0.6931471,  and 
thence  we  get  log  2  =  the  common  logarithm  of  2,  equals 

^'?^^^itJ^  =  0.6931471  X  0.4342944  =  0.3010299, 
log  10 

which  agrees  with  the  common  logarithm  of  2,  as  given  by 

the  logarithmic  tables.     Eeciprocally, 

log.  2  X  log  10  =  log  2  --  0.4342944  =  0.6931471, 

equals  the  hyperbolic  logarithm  of  2. 

It  follows  from  what  has  been  done,  that  the  calculation 


LOGARITHMIC   FORMULA.  63 

of  a  table  of  logarithms  to  any  base  may  be  considered  as 
being  reduced  to  the  calculation  of  hyperbolic  logarithms. 
For  examples  in  illustration  of  the  calculation  and  use  of 
logarithms  in  the  solution  of  problems,  the  reader  is  referred 

to  p.  527,  &c.,  of  my  Algebra. 

.     Eesuming  e'^^l+ajH-—  +  ——  +,  &c.,  from  {})"\ 

p.  51,  and  changing  x  successively  into  xV—1  and  —  xV—i^ 
we  get  the  equations 

aj2       a^yHT 


_      _  ^       _^ x' 

~  ^       1.2  +  1.2.3.4      1.2.3.4.5.6  +'  ^^•' 

"•- 12:3-^  12:3x5 -'H^-^> 

^  ~  1.2  ^  1.2.3.4       1.2.3.4.5.6  ^'       ' 


(^  -  r2j  + 1:2:8.4:5 -'H"^-'' 


By  taking  the  half  sum  and  half  difference  of  these  equa- 
tions, we  get 

(^v^+  ,-x»'^)  ^2  =  1-   ^  +  j^^  -,   &c,     and 

(.^---  .---)  ^  2 1/^  =  1  -  j^  4-  ^^  -,  &o. ; 

which  (in  trigonometry)  are  called  the  cosine  and  .s{??e  of  a?. 

Denoting  the  sine  and  cosine  by  writing  sin  and  cos  for 
them,  the  preceding  equations  may  be  written  in  the  forms 

sm  X  — ^=^- ,  and  cos  x  = . .  (c). 


54  DIFFERENTIATING  LOGARITHMS. 

By  adding  the  squares  of  (c'),  we  get  sin-  x  +  cos'  a?  =  1  ; 
wliicli  is  also  evident  from 

sin  a?  =  a?  —  +,  &c.,  and  cos  a?  =?  1  —  r-^  +,  &c. 

We  are  now  prepared  to  show  how  to  find  the  differentials 
of  logarithmic,  exponential,  and  circular  functions. 

(4.)  To  show  how  to  find  the  differentials  of  logarithmic 
and  exponential  functions. 

We  will  show  how  to  find  the  differential  of  a  variable  or 
function  represented  by  log  x. 

From  (a),  given  at  p.  51,  if  we  put  a?  for  1  +  a^  we  must 
clearly  put  a?  —  1  for  a,  and  we  shall  have 
,  ^  (aj-l)2       (x--rf       (x-Vf        „    , 

in  which  m  is  the  modulus;   consequently,  by  taking  the 
differential  of  this,  m  being  constant,  we  shall  have 
d([ogx)  =  m\l-{x-l)  4-  {x-Vf  -{x-lj  +,kQ.'\  x  dx 

mdx        7ndx 

~  l'V{x  -  1)  ~  ~^  ' 

dx 
and  when  7n  =  1,  we  have  d  (log  x)  =  — . 

X 

Hence  the  differential  of  the  logarithm  of  a  variable  or 
function  can  be  found  by  the  following 

BULK 

1.  Divide  the  differential  of  the  variable  or  function  by 
the  variable  or  function,  and  the  quotient,  multiplied  by  the 
modulus,  gives  the  differential. 

2.  If  the  modulus  is  unity,  or  the  logarithm  hj^perbolic, 
then  divide  the  differential  by  the  variable  or  function,  for 
the  differential 


EXAMPLES.  65 

Eemabks. — 1.  Wlien  it  is  possible,  hyperbolic  logaritbins 
ought  always  to  be  used  in  finding  differentials,  because 
their  differentials  are  of  the  most  simple  forms. 

2.  It  clearly  results  from  the  rule  that  the  differential  of  a 
variable  or  function  equals  the  differential  of  its  hyperbolic 
logarithm  multiplied  by  the  variable  or  function. 

EXAMPLES. 

1.  The  differentials  of  log  (a  +  x)  and  log  ax=  log  a  -f  log  x. 

mdx        ,  7n.dx 

are and . 

a  +  X  X 

cc 

'     2.  The  differentials  of  log  {x  -f  y)  and  log  -  =  log  x  —  log  y, 

dx  +  dy      ,  dx       dy       ydx  —  xdy 

are and -  = . 

X  -\-  y  X         y  xy 

3.  The  differentials  of  log  {a?  +  x") 

and  log  (rt^  —  a?^)  =  log  (a  +  x)  +  log  {a  —  x\ 

27nxdx  T         dx  dx  2xdx 

are      -7- — 5      and      — ; = ^ ^ . 

a'  -\-  x""  a  -\-  X       a  —  X  w  —  xr 

4.  The  differentials  of 

log  (ar  —  a^)  =  log  (a?  +  a)  -f-  log  (a?  —  a) 

2 
and  log  -  —  log  2  —  log  a?, 

a? 

d\c  dx  2xdx 

are  h 


X  +  a      X  —  a      a^  —  (T^ 
which  is  the  same  as  to  divide  d  (a>^  —  a-)  by  (a^  —  a-),  and 
dx 


0.  The  differentials  of  log  |/  (a^  +  a?')"*'  ==  log  {a^  +  ar*)  ^  and 
log[^+V(^±a^)]are^;^, 


V{^±a^' 


66  DIFFERENTIATING  EXPONENTIALS. 

6.  The  differential  of  log  ^jfl  "^  ^-"^,  is 

xdx        /I  1  \  _       ^^^ 

^{a^  +  a?-)  \  |/(tt*  +  ar^)  — a  ~  |/(a»  +  ar)  +  a/  ~  a?|/(a'  +  aj^) 

7.  The  differential  of  ax""  is  aa?"*  x =  maa?"*  -  Wa?. 

a? 

8.  The  differential  of  xy  is 

xy  X  d  log  ajj^  =  a^ycZ  (log  a;  +  log  y) 

=  a?y  ( h  —1  =  ydx  -f  xdy. 

9.  The  differential  of  ^,  is?(^  -  ^)  =  IH^ZI^. 

y'      y\x        yJ  v" 

10.  The  differential  of  a^  is 

a^  X  tZ(loga^)  =  a^d(\oga  x  x)  =  a^ loga  x  dx ; 

which    can    be    also    found    from    assuming    y  =  a*,    or 

diJ 
log  y  =  x  log  a,  or  —  =  log  adx,  or  dy=^y  log  adx=a^  log  ac?a?, 

as  before. 

It  is  hence  evident,  that  when  the '  exponent  of  an  expo- 
nential is  alone  variable,  we  can  find  the  differential  of  the 
exponential  by  the  following 

RULE. 

Multiply  the  hyperbolic  logarithm  of  the  base  or  root  of 
the  exponential  by  the  exponential,  and  the  product  by  the 
differential  of  the  variable  exponent 

Remark. — If  the  base  of  the  exponential  is  also  variable, 
then  we  must  add  the  differential,  regarding  the  base  as  alone 
variable  to  the  preceding  differential ;  and  the  result  will  be 
the  complete  differential,  when  the  base  and  exponent  of  the 
exponential  are  vaiiable. 


DIFFERENTIATING   EXPONENTIALS.  57 

EXAMPLES. 

1.  The  differentials  of  2^  and  3*^,  are  2^  x  log  2  x  dx^  and 
3^  log  S  X  dy:  noticing  that  2  and  3  are  the  constant  roots  of 
the  exponentials,  whose  variable  exponents  are  x  and  y. 

2.  The  differentials  of  e"^  and  e~%  are  e'^dx  £ind  —  e~^dx] 
since  log  e  =  l. 

3.  The  differentials  of  Ja^  and  c'''^,  are  ha^  log  a  x  dxj  and 
c'^^logc  X  ao?a7. 

4.  The  differentials  of  e^°s-^  and  a^°sx^  are 

^iogx_^   and   a'^^^logax  — . 
X  °  X 

5.  The  differentials  of  ay"^  and  2/-^,  are 

xay'^-^dy  +  ay"^  log  y  x  dx 
and  —  xy~^~^dy  —  y~^  log  2/<^a?, 

as  is  clear  from  the  rule  and  remark. 

6.  The  differentials  of  a^'  and  e"',  are  a*' log  ae^c^a;  and 
e°'  log  a  X  a^dx ;  noticing  that  e'^  and  a^  are  variable  expo- 
nents of  a  and  6,  and  that  e  stands  for  the  hyperbolic  base. 

7.  The  differentials  of  z""'  and  (log  x)  '''^  ^,  are 

s""  log 2  X  {yx'-^  -^  dx  -\-  x^  log  ajc/^/)  +  x^z"  -^ 6/5;, 
and  (log  a;)^"«^  x  log  (log  x) \-\ogx  x  (log  a?)^°e^-^  x  — 

=  [logxy^^ ^  X  (1  +  log^^')  —  : 

X 

noticing,  that  the  notation  log"  x  is  used  for  log  (log  a?), 
and  we  may  also  represent  log  [log  (log  a?)]  by  writing  log^  x ; 
and  so  on,  to  any  extent. 

8.  The  differentials  of  e  v(« -x')  and  e  ''^^'* ,  are 

3* 


68  CIRCULAR  FUNCTIONS. 


and  e'°«*~-^ . 

X  log  a; 

9.  The  differentials  of  e^"^-^  and  ^-^  ^^ ,  are 


^xi^^T^   |/Zri,y;c  and  e-^ "^^^  y.  —  \^  —  V  dx . 

10.  The  differentials  of  a-^  ^~^  and  a"^  ^'-^ ,  are 
^i  v^  1  iQg  ^  X  ^/^  1^—1,  and  rt-''  ^"-^  log  «  X  —  dx  V  —  1. 

(5.)  TFi?  i^e7^  noz^  5/io>w;  Aoi<?  to  find  the  differentials  of  cir 
cular  functions. 


From 
and 

{c)  page  53,  we 
cos  a?  = 

have  sin  x 

e"  ^^^  +  e- 

g.T 

^_,-x^in 

xvn 

2V  -1 

'?, 

J 

whose  differentials  give 

a  (sin  x)  = 

-'  +  e- 
2 

-xV~ 

i 
-dx  : 

=  cos  a;<^a?, 

<?(cos  x)  = 

,xVZT 

—  (5- 

.xV~) 

-  X    V  -idx 

2 

.  .-^^- 

.,-x^: 

^_ 

-  sinxdx. 

2  1/^1 

By  adding  the  squares  of  these  differentials,  we  shall  get 
{d  sin  x)-  +  {d  cos  x)-  =  cos^  a?^ic^  +  sin^  xdixr^ 

=  (cos^a?  +  sin-  x)  da^  =  doi^j 
since  we  have  shown,  at  page  54,  that  cos^a?  +  sin^aj  =  1. 

Remark. — It  is  clear  from  the  expressions  for  sin  x  and 
cos  a?,  that  they,  together  with  x  and  dx,  represent  numbers 
or  geometrical  ratios,  and  not  quantities. 

It  clearly  follows  from  what  has  been  done,  that  we  can 
find  the  differentials  of  the  sine  and  cosine  of  any  variable 
by  the  following 


DIFFERENTlATINa  SINES,   ETC.  59 

RULES. 

1.  The  differential  of  the  sine  equals  the  cosine  multiplied 
bj  the  differential  of  the  variable. 

2.  The  differential  of  the  cosine  equals  minus  the  product 
of  the  sine  and  the  differential  of  the  variable. 

EXAMPLES. 

1.  The  differentials  of  sin  2ic  and  cos  2x,  are  2  cos  2xdx 
and   —  2  sin  2xdx. 

2.  The  differentials  of  sin  7nx  and  cos  mx,  are 

m  cos  ?nxdx    and     —  m  sin  mxdx. 

3.  The  differentials  of  sin  (a  ±  x)  and  cos  (a  ±  x),  are 

±  cos  {a  ±  aj)  c/a?    and     ^  sin  (a  i  x)  dx. 

4.  The  differentials  of  sin^  x  and  cos'^  x ,  are 

2  sin  X  cos  cct^a?     and     —  2  sin  a?  cos  xdx. 
6.  The  differentials  of  sin"'  x  and  cos"*  x  are 


noticing,  that  the  exponent  m  denotes  the  mth  powers  of 
sin  X  and  cos  x. 

6.  The  differentials  of  tan  x  = and  cot  x  =:  -r^ — ,  are 

cos  a?  sma?  ' 

-  ^  d  s'mx  X  cos  X  —  d  cos  a?  x  sin  a? 

a  tan  x  = z 

cos- a? 

cos^xdx  4-  sin^  a?r/aj  dx 


=  sec^  xdic 
COS"  X  cos-  X 

and        d  cot  a?  = 


dx  „ 

^-T—  =  —  cosec-  xdx. 


60  DIFFERENTIATING  TANGENTS,   ETC. 

Hence,  the  differential  of  the  tangent  of  a  variable^  equals 
the  differential  of  the  variable  divided  by  the  square  of  its 
cosine  or  multiplied  by  the  square  of  its  secant  /  since  unity 
divided  by  the  cositie  is  (^Vl  Trigonometry)  called  the  secant. 

And  the  differential  of  the  cotangent  of  a  variable,  equals 
tninus  the  differential  of  the  variable  divided  by  the  square 
of  its  sine  or  multiplied  by  the  square  of  its  cosecant. 

EXAMPLES. 

1.  The  differentials  of  tan  2.»  and  cot  2.^?,  are 

%lx  ,  %lx 

and 


cos^  ^x  sin'-^  ""Ix  ' 

2.  The  differentials  of  tan  mx  and  cot  mx^  are 

mdx  ,  mdx 

cos"^  rax  sin^  mx  ' 

3.  The  differentials  of  tan  {a  ±  x)  and  cot  {a  ±  x),  are 

±dx  ,  _         dx 

and  ^ 


cos^  (a  ±  x)  sin^  {a  ±  x) 

4  The  differentials  of  tan  x"^  and  cot  a?*" ,  are 

7?iisinx^~'^dx        ,  m  cot  x'^  ~  ^  dx 

r and r—^ 

cos^  X  sm^  X 

5.  It  is  easy  to  perceive  that  we  may,  in  much  the  same 

way,  find  the  differentials  of and  — —  ;  which  edve 

•^  cos  X         sm  a?  ° 

,      1              d  cos  X      sin  xdx  , 

d  = X —  = 5 —  =  tana?  sec  xdx ; 

cos  X  COS^  X  COS^  X 

and  in  like  manner 

,1  dsinx  ^  , 

a = r-^—  =  —  cot  X  cosec  xdx. 

sm  X  sm^  X 

Because and  -: are  called  the  secant  and  cosecant 

cos  a?  sm  a? 


DIFFERENTIATING  MODIFIED   FUNCTIONS.  61 

of  .'»,  we  hence  find  the  differentials  of  the  secant  and  co- 
secant of  any  variable,  by  the  following 

RULE. 

The  differential  of  the  secant  of  a  variahle  equals  the 
product  of  the  tangent,  secant,  and  differential  of  the 
variahle. 

And,  the  differential  of  the  cosecant  of  a  variable  equals 
minus  the  product  of  the  cotangent,  cosecant,  and  differen- 
tial of  the  variable. 

Thns,  the  differentials  of  sec"'  x  and  cosec""  x,  are 

tan  X  sec  u?  x  m  ^QQ!^~^xdx 

and  —  cot  x  cosec  x  x  m  cosec""  ~^  dx', 

and  the  differentials  of  sec  {aP"-  4-  ^"^)  and  cosec  {al^  —  x^\  are 

tan  {or  +  x"')  sec  (a"'  +  a?"')  x  mx"^-^  dx, 

—  cot  {or  —  »"*)  cosec  {oT  —  x"^)  x  m^-^dx. 

6.  Because  versin  a?— 1  —  cos  x  and  coversin  a?  =  1—  sin  x, 
their  differentials  are  sin  xdx  and  —  cos  xdx ;  which  are  the 
same  as  those  of  the  cosine  and  sine  after  their  signs  are 
changed. 

7.  Since  snversine  of  a?  =  1  -f  cos  x^  and  cosuversine 
a?  =  1  +  sin  a?,  their  differentials  are  —  sin  xdx  and  cos  xdx  ; 
which  are  the  same  as.  those  of  the  cosine  and  sine. 

8.  The  differentials  of  sin  (sin  a?)  and  cos  (sin  x\  are  evi- 
dently d  sin  (sin  x)  =  cos  (sin  x)  cos  xdx, 

and  d  cos  (sin  x)  —  —  sin  (sin  x)  cos  xdx ; 

and  the  differentials  of  sin  (cos  x)  and  cos  (cos  x),  are 

d  sin  (cos  a?)  ==  —  cos  (cos  x)  sin  xdx, 
and  d  cos  (cos  x)  =  sin  (cos  a?)  sin  xdx  ; 

and  so  on,  for  other  analogous  forms. 


62  DIFFERENTIATING  MODIFIED  FUNCTIONS. 

9.  The  differentials  of  log  sin  x  and  log  cos  jc,  are 

,        .  c^sina; 

a  looj  sin  X  =  — r =  cot  xdx, 

°  sm  ic 

and  d  log  cos  x  =  =  —  tan  xdx ; 

cos  X 

and  these  multiplied  by  the  modulus  (//i) ,  will  give  the  dif- 
ferentials of  log  sin  X  and  log  cos  x. 

10.  The  differentials  of  log  tan  x  and  log  cot  a?,  are 

,   ,      ,  d.  tan  X  dx  dx  2dx 

c? .  log  tan  a;  =  — =  — -, — =  -. =  — — tt  » 

°  tan  X        cos-  x  tan  x      sm  x  cos  a?      sm  2x 

d.  cot  a?  dx  2  Ix 


and    c? .  loff  cot  a? 


and  we  may  proceed  in  like  manner  in  all  analogous  cases. 

(6.)  Since,  to  find  the  preceding  values,  it  is  necessary  to 
know  those  of  sin  a?,  cos  x,  tan  x,  &c.,  when  x  and  dx  are 
given ;  we  will  now  show  how  to  obtain  their  values  to  any 
degree  of  exactness  that  may  be  required,  by  converging 
series. 

To  the  end  in  view,  we  will  find  the  expansions  of 
sin  (a?  ±  A)  and  cos  (a?  ±  A),  when  arranged  according  to  the 
ascending  powers  of  h. 

Thus,  if  we  put  sin  a?  for  X,  and  sin  (a?  ±  //)  for  X',  and 
±  h  for  A,  in  Taylor's  Theorem,  or  (a),  given  at  p.  16,  we 

,    ,.  ,         c?X      c^sina?  d^X.      dcoax 

sliali  nave  -3—  =  — -. —  =  cos  x,    -j-^  =  — -. — -  =  —  sm  a?, 

d'X  d'X   . 

~T^  =  —  cos  X,    -r-T-  sm  X,  and  so  on. 

djcf  dx*  ' 

Hence,  from  the  substitution  of  the  preceding  values  in 
(a),  we  get,  after  duly  ordering  the  terms, 


EXPANSIONS  OF  SIN  (x  ±  h)  AND   COS  {x  ±  h).         63 
Sin  {x±h)  =  sm  x  il  —  -^  +  12^3  4  ~'  *^^- J 

=  sin  X  cos  h  zt  cos  x  sin  h  (see  p.  53)  ....  ((/). 
In  like  manner,  we  easily  get 

cos  {x  ±  h)  =  cos  a?  ^1  -  ^  +  ^^  -,  &c.  j 

=^^^^^(^-il3  +  i:2:S45-'H 

—  cos  x  COS  A  ^  sin  a?  sin  A  (p.  53)  .  .  .  .  {d'). 

If  we  put  A  =  a?,  and  use  the  upper  signs  in  these  formu- 
lae, they  give 

sin  2^c  =  2  sin  x  cos  i»,     and    cos  2x  =  cos^  x  —  sin"  a; ; 

or,  since  sin-  x  +  cos-  a?  =  1,  we  have  cos^  a;  =  1  —  sin^  a?, 
which  reduces 

cos  2x  =  COS"  a?  —  sin^  x    to     cos  2a?  =  1  —  2  sin^  x  ; 
which,  by  changing  x  into  ^,  becomes  cos  a?  =  1  —  2  sin^  ^. 

As  an  example  of  the  use  of  the  last  of  these  formulae, 
we  shall  successively  put  a;  =  1.5  and  1.6,  and  thence  get 

-  z=  0.75;  and  -  —  0.8,  for    the   corresponding   values   of 

X  -^2. 

From  the  substitution  of  these  values  in 

.XX        \2/  V2/ 

^•"2  =  2 -1X3  +1:2:81:5-  '^°-' 
we  shall  get 


64     REPRESENTATIONS  OF  1,  SIN  a?,  AND  COS  X. 

sin  0.75  =  0.75  -  0.070312  +  0.001977  -  0.000026  +,  &c., 

=  0.681639, 
and 

sin  0.8  =  0.8  -  0.085333  +  0.002730  -  0.000041  +,  &c., 
=  0.717356. 

Hence,  we  get    2  sin-  |  =  2  sin^  0.75  =  0.929268,   and 
thence  we  have 

cos  X  —  cos  1.5  =  1  —  0.929268  =  0.070732. 
In  a  similar  way,  we  have  2  sin-  0.8  =  1.029059,  and  thence 

cos  1.6  =  1  -  1.029059  =  -  0.029059. 
Hence,  there  is  clearly  a  value  of  x  greater  than  1.5  and  less 
than  1.6,  which  we  shall  represent  by  -,  such  that  we  shall 

have  cos  -  =  0 ;  and  thence  from  cos-  ^  +  sin-  -  =  1,  we 
get  sin*  -  =  1,     or    sin  -  =  ±  1. 


&c. 


From  COS. =  l-2sin=|  =  l-2'(|)-A|A+, 

by  putting  a?  =  0,  we  also  have  cos  0  =  1,  and  sin  ^  =  0, 
or  sin  a?  =  0 ;  and  from 

cos  TT  =  oos^  ^  —  sin- ,.,  since  cos^  o  =  ^j  ^^^  s^^^o  =  1»  ■ 
we  have  cos  tt  =  —  1. 

(7.)  For  convenience  in  what  is  to  follow,  we  now  propose 
to  show  how  to  represent  1,  sin  a?,  and  cos  a?,  geometncally. 

Since  sin^  x  +  cos^  a?  =  1,  it  is  clear  that  the  sum  of  no 
two  of  the  three,  1,  sin  a?,  cos  x^  can  be  less  than  the  third, 
while  their  difference  can  not  be  greater  than  the  third. 


KEPRESENTATIONS   OF   1, 


65 


These  results  correspond  to  well-known  properties  of  the 
sides  of  a  rectilineal  triangle;  viz.,  that  the  sum  of  any  two 
of  its  sides  can  not  be  less  than  the  third,  while  their  differ- 
ence can  not  be  gi'eater ;  properties,  that  evidently  follow 
from  the  consideration,  that  a  straight  line  is  the  shortest 
distance  between  its  extremities. 


Thus,  let  ABC  denote  a  rectilineal  triangle ;  such,  that  AC 
equals  any  unit  of  length,  while  CB  and  AB  are  the  same 
parts  of  AC  that  sin  x  and  cos  x  are  of  1 ;  then,  it  is  clear 
that  AC,  CB,  and  AB,  may  be  taken  as  representatives  of 
1,  sin  a?,  and  cos  x.  Similarly,  if  we  take  the  equation 
sin^  x  +  cos^  a?'  =  1,  such  that  sin  x'  is  the  same  part  of  1 
that  AB  is  of  AC,  it  is  manifest  that  1,  sin  x\  and  cos  x\ 
will  also  be  represented  by  AC,  AB,  and  BC ;  consequently, 
sin  X  ==  cos  x'^  and  cos  x  =  sin  x\ 

From  {d)^  page  63,  sin  (x  +  h)  =  sin  x  cos  h  +  cos  x  sin  h ; 
which,  by  putting  x'  for  A,  becomes 

sin  {x  +  x')  =  sin  x  cos  x'  +  cos  x  sin  x'  =  sin^  x  -f  cos^  x  —  1', 
since  cos  x'  =  sin  x  and  sin  x'  =  cos  x,  and  that 

sin  ^  a?  +  cos  ^  x=^l. 
Because   sin  (a?  +  a?^)  =  1,  and  that  at  page   64  we  have 

sin  ^  =  1 ,  it  follows  that  we  must  have  a?  +  a?'  =  ^ ,  in  which 


66  REPRESENTATIOXS   OF   1,    SIN  27,    AND   COS  3?. 

^  is  a  value  of  x  +  x'  that  lies  between  1.5  and  1.6.     (See 
page  64.) 
Because  (page  53)  C  B  =  sin  a;  =  a?  —  — — -  -\-        '  ~,  &c., 

x'^  x'^ 

and  that  AB  =  since'  =  a?'—  —^-^  -h  ^^ir-r-^,  &;c., 

by  adding  these  we  have 

CB  +  AB  =  sin  a?  +  sin  x' 

i^  +  x'\    x'  +  x"        . 

=^  +  ^- -1:2:3- +  i:2j5.-i:5-'^^- 

=  {w  +  x)\^ TO"-  +'^-)' 

which  is  greater  than  the  side  AC ;  and  if  a?  =:  0,  sin  x'  =  AC, 
or  if  a;'  =  0,  sin  x  =  AC.     Since 

AC  =  1  =  sin(;i.+  x')  ={x  +  x)  (l  -  ^±M±^+,&c.), 
we  hence  get 

,        ,^  r      x"  —  xx'  ^x""       ,    \ 

^   /  „  /,        a?"  +  2a'a;'  -^  x""        ,     \ 

It  hence  follows  that  x  and  x'  must  be  represented  by  the 
angles  A  and  C  ;  for  if  sin  a?  =  0  we  have  x  and  the  angle  A 
each  equal  to  naught,  and  AC  coincides  with  AB ;  and  in 
like  manner  if  sin  x'  =  0,  AC  coincides  with  BC.  Because 
of  the  inequality  AC  -f  BC  >  AB  ,  if  X  represents  the  angle 
B,  it  is  clear  that  we  must  have 

X  +  a.-  -^3-  +,&c. 
^(X  +  .)(l--^^-y3^-%,&c.)>.--^4-,&a 


X  =  X   -\-  x'  =  A  RIGHT  ANGLE.  67 

for  the  proper  representation  of  the  inequality;  consequently, 
AC  being  expressed  in  terms  of  X  in  a  way  similar  to  the 
representations  of  the  other  sides  in  terms  of  their  opposite 
angles,  it  clearly  follows  that  AC  must  be  the  sine  of 
X  =  sin  ABC. 

From  AC  =  xi-x'-  ^^jj^  +,  &c.,  -  X-  j^  +  ,  &c, 

we  must  have  X  or  the  angle  ABC  equal  to  a?  +  x\  the  sum 
of  the  angles  A  and  C .  Hence  (see  figure),  if  from  the 
right  angle  Bf  we  draw  the  right  line  BD  meeting  AC  in  D, 
so  as  to  make  the  angle  CBD  =  the  angle  C,  we  shall  have 
the  anoxic  ABD  =  the  ansjle  A . 

Hence,  the  lines  AD,  DB,  and  DC,  are  equal,  and  the  points 
A,  B,  C,  lie  in  the  circumference  of  a  circle  whose  center  is 
D  and  radius  DB .  If  the  angles  A  and  C  equal  each  other, 
it  is  clear  that  AB  =  BC,  and  of  course  AC'  =  AB^  +  BC^ 
or  4AD^'=2AB=  or  AB- =.  2AD- =  AD^- +  BD^;  con- 
sequently, in  the  triangle  ADB  the  angle  D  equals  the  sum 
of  the  remaining  angles  of  the  triangle.  But,  since  the 
triangles  ADB  and  CDB  are  clearly  identical,  it  results  that 
their  angles  at  D  must  equal  each  other,  and  of  course  from 
the  well-known  definition  of  a  right  angle,  each  of  them  is  a 
right  angle.  Hence,  the  angles  at  A  and  B  in  the  trian- 
gle ADB  are  together  equivalent  to  a  right  angle ;  and  in  the 
triangle  CDB,  the  sum  of  the  angles  at  C  and  B  is  equiva- 
lent to  a  right  angle.  Hence,  the  sum  of  the  angles  of  the 
triangle  ABC  is  equivalent  to  two  right  angles,  and  because 
the  angle  B  equals  the  sum  of  the  angles  A  and  C,  it  is  clear 
that  B  is  a  right  angle,  and  that  the  sum  of  the  angles  A  and 
C  is  equal  to  a  right  angle ;  and  because  the  angles  A  and  C 
make  the  same  sum,  whether  they  are  equal  or  unequal,  it 
clearly  follows  that  their  sum  is  always  a  right  angle. 


68 


THE  ANGLE   IN  A  SEMICIRCLE   IS   RIGHT. 


Also,  because  the  angle  B  is  always  in  a  semicircle  whose 
center  is  D  and  diameter  AC,  it  follows  that  the  angle 
inscribed  in  a  semicircle  is  always  a  right  angle. 

Eeciprocally,  if  one  angle  of  a  triangle  is  right,  the  sum 
of  the  other  two  angles  is  right,  and  the  square  of  the  numer- 
ical value  of  the  side  opposite  to  the  right  angle  equals  the 
sum  of  the  squares  of  the  numerical  values  of  the  other  two 
sides.  For  ABC  (see  fig.)  being  the  triangle,  a  circle  de- 
scribed on  AC  as  a  diameter,  must  evidently  pass  through 
the  right  angle,  and  the  triangle  coincides  with  one  of  the 
triangles  that  have  been  considered ;  and  thence  the  truth  of 
the  proposition  is  manifest.  It  may  be  added  that  the  sum  of 
the  three  angles  of  any  rectilineal  triangle  is  easily  shown  to 
be  equal  to  two  right  angles. 

(8.)  We  now  propose  to  ^how  how  to  find  the  numerical 
values  of  angles. 


Eesuming  the  right  triangle  ABC  from  p.  65,  we  have, 
according  to  what  is  there  supposed,  AC  to  represent  any 
arbitrary  unit  of  length,  while  the  angles  A  and  C  are 
represented  by  a?  and  x\  and  CB  =  sin  a?,  AB  ==  sin  x' 
=  cos  X.  If  from  A  as  a  center,  with  AC  as  ^a  radius, 
the  arc  CGr  is  described  meeting  AB  produced  in  G,  it  will 
represent  the  value  of  sin  x.    By  taldng  the  differentials  of 


X  —  THE   LENGTH   OF   THE   ARC  GC.  69 

CB  =  sin  X  and  AB  =  cos  x^  we  sliall  (as  at  p.  61)  have 
d.  CB  =  cos  xdx  and  d.  AB  =  —  sin  xdx^  which  give  (as  at 
p.  58)  \\{d.  CB)"^  +  (t/.  AB)-]  =  cfe,  supposing  x  and  sin  x 
to  increase  while  cos  x  decreases.  If  from  B  toward  A, 
BH  is  set  off  to  represent  d ,  AB  ==  —  sin  xdx  and  HE 
drawn  parallel  to  CB,  meeting  the  tangent  to  the  arc  CGr  at 
C  in  E ;  and  if  through  C,  CD  is  drawn  parallel  to  AB, 
meeting  HE  in  D ;  then,  EC  represents  dx^  and  ED  =:  o^.CB 
=  cos  xdx.  For  the  right  triangles  ACB  and  ECD  give 
the  proportions 

AC  or  1 :  EC  ::  cos  x  :  ED,  and  1 :  EG  ::  sin  a; :  CD  =  BH, 

which  give    DE  =  EC  x  cos  a?,  and  BH  =  EC  x  sin  x. 

Since  (neglecting  the  signs)  BH  =  sin  xdx^  the  second  of 
these  equations  gives  EC  x  sin  a?  =  sin  xdx^  or  dx  =  EC ; 
consequently,  the  first  becomes  DE  =  cos  xdx,  as  it  ought 
to  be.  Because  the  arc  GC  and  the  angle  x  commence  to- 
gether at  G,  and  increase  together  from  G  toward  C,  and 
that  the  increase  of  the  arc  at  any  point  is  clearly  in  the 
direction  of  the  tangent  (at  the  point),  CE  evidently  repre- 
sents the  differential  of  the  arc  GC ;  consequently,  since 
dx  —  GE,  it  follows  that  dx  represents  the  differential  of  the 
arc  GC,  and,  of  course,  x  equals  GC ;  agreeably  to  what  has 
been  supposed. 

Eemarks. — 1.  It  is  easy  to  perceive  that  we  may  proceed 
in  much  the  same  way  as  above,  to  find  the  differential  of  any 
proposed  arc  of  any  plane  curve,  by  expressing  it,  in  terms 
of  the  differentials  of  its  rectangular  co-ordinates,  like  AB 
and  CB ;  that  is, .  by  taking  the  square  root  of  the  sum  of 
the  squares  of  their  differentials  at  any  point  of  the  curve, 
for  the  differential  of  the  curve  at  the  same  point. 

2.  In  our  reasonings  we  have,  and  shall,  generally,  take  it 


70  FINDING  THE  ARC  X   IN  TERMS   OF   t. 

for  granted  that  the  reader  is  familiar  with  the  definitions 
and  leading  principles  of  Geometry  and  Trigonometry. 
Thus,  in  the  figure,  supposed  to  be  constructed,  AC,  CB, 
AB,  are  called  the  radius,  sine,  and  cosine  of  the  arc  GC ; 
also,  AG,  GF,  and  AF,  are  called  the  radius,  tangent,  and 
secant  of  the  same  arc. 

3.  AC  being  represented  by  1,  since  the  equiangular 
triangles  ABC  and  AGF  give  the  proportion 

AB  :  BC  ::  AG  :  GF  =  t,     or    cos  x  :  sin  x\\\\t, 
or  its  equivalent,  sin  x  =■  t  cos  aj, 

in  which  t  =  the  tangent  of  x.     Since 

sin  a;  =  a?  -  ^-^-g,  &c.,    cos  a?  =  1  -^  -f-  ^^a  ~'  ^'^' ' 

consequently,  the  preceding  equation  may  be  written  in  the 
form, 

di?  m^  i         0?  x^  \ 

"^ - 12:3  +  risAS -'*"•  =  n^ - 1.2  +  ixsii -  *°) ' 

which  clearly  shows  that  x  can  be  expressed  in  a  series  of 
the  odd  integral  powers  of  t. 

For  a  simple  inspection  of  the  terms  shows  t  to  be  the 
first  term  of  the  series ;  and  to  get  the  second  term,  we  put 
t  +  A^'  for  aj,  and  thence  have 

^  +  ^^'~  m  +'  &c.  =  25  -  ^  +,  &c. ; 
consequently,  if  we  determine  A,  on  the  supposition  that  the 
terms  involving  f  destroy  each  other,  we  shall  have 

A/3         ^'     _         ^  .   _         1  1      _        1 

^^~iM-~i:%'  ""^  ^-"1:2  +  1:23- "a: 

If  ^  —  -  +  Af  is  put  for  X,  we  shall  in  like  manner  get  A  =  ^ ; 

o  O 

and  so  on.    Hence,  we  shall  have 


FINDING  THE   CIRCUMFERENCE   OF  A  CIRCLE.  71 

«;=<-f +  |-y+,&c (.); 

whicli  is  a  very  useful  formula  for  finding  the  circumference 

of  a  circle. 

l^lius,  if  X  is  the  numerical  value  of  half  a  right  angle, 

since  X  ■=.  x\  we  have 

sin  3? 

sm  X  =  cos  X,  and,  oi  course,  t  =  =  1 ; 

'  '  cos  iC 

consequently  (since  ^  expresses  the  numerical  value  of  a 

right  angle),  by  putting  1  for  t  and  -  for  x,  we  shall  have 
77        ,        1        1        1 

Again,  if  x  is  one-third  of  a  right  angle,  we  shall  have 
x'  ==  2x;     and    cos  x  =  sin  x'  =  sin  2a;  =  2  sin  x  cos  Xj 

or  sin  a?  =  -,     and  thence     cos  x  =    -^'y 

L  it 

consequently,  from     t  =  we  get     ^  =  -— 

cos  X  yo 

From  the  substitution  of  this  value  of  t  in  (e),  we  clearly 

.    ^  i    /i         11  1  1  P    \ 

^^'    6  ^  ^3  1^  -  O  +  5:3^-  773^3  +  9:3^-'  ^^-j' 

which  will  enable  us  to  find  the  numerical  value  of  rr  to  any 

required  degree  of  exactness. 

The  value  of  tt  to  eight  decimal  places  is  easily  found  to 

be  3.14159265 ;  which  is  clearly  the  numerical  value  of  two 

right  angles,  or  the   semicircumference  of  a  circle  whose 

radius  is  the  unit  of  length ;  consequently,  the  product  of 

TT  and  R,  the  radius  of  any  other  circle,  gives  Rtt  for  the 

length  of  the  semicircumference  of  the  circle  whose  radius 

is  B. 


72  IMPLICIT  FUNCTIONS  OF 

For  series  of  more  rapid  convergency  than  the  above,  the 
student  is  referred  to  page  70,  volume  1,  of  Lacroix's  "  Calcul 
Ditfei-entiel,"  and  to  page  797  of  Eutherford's  edition  of 
"  Button's  Mathematics." 

(9.)  We  will  now  show  how  to  find  the  differential  of  an 
arc  regarded  as  a  function  of  its  sine,  cosine,  etc.;  which  are 
sometimes  called  inverse  functions. 

1.  If  sin  2  =  y  and  cos  s  =  aj,  we  get  from  what  is  done 

at  page  58,  cos  zdz  =  dy    and    sin  zdz  =  —  dx,  or 

/  •         •  c             o        -.N  7             dy           ,   ,                dx 
(smce  sm-  z  +  cos-  z=i)dz=  — ~z=:r  and  dz  = ■ ; 

Vl-f  VT^' 

and  in  like  manner,  if  we  put  tan  z  =  t   and  cot  z  =  t' {  we 
get,  from  page  59, 

— —   =  dt     and     .  „    =  —  dt'  \ 


or  dz  =  co^zdt  and     dz=  —  &\v^zdt', 

which  are  equivalent  to 

dt  ,     ,  dt' 

dz  =  ■ :z  and  dz 


1  +  2?     1-^t"' 

Also,  if  sec  s  —  5  and  cosec  z  =  s\  we  get,  from  what  is 
shown  at  page  61, 

tan  z  sec  zdz  =  ds    and    cot  z  cosec  zdz  =  —  ds' ; 


which,  from  tan  z  =  Vs^  —  1  and    cot  z  =   Vs'^  —  1,  are 

reducible  to 

,               ds  ,  ,                    ds^ 

dz  =  — —  and  dz 


In  much  the  same  way,  from  page  61,  if  we  put 
versin  z  =  l—-  cos  z  =  v    and    coversin  z  =1  —  sinz  =  v\ 
we  get 

sin  zdz  =  dv    and    cos  zdz  =  —dv\ 


ARCS   DIFFERENTIATED.  73 

,           dv             1       ,                dv' 
or  dz  =  — —     and    dz=  —  : 

sin  3  cos  s 

consequently,  since  cos  s  =  1  —  -y  and  sin  s  =  1  —  -y',  we  get 

,  dv  T      .  dv 

dz  = J    and     dz=  — 


V2v'  -  v"' 

2.  It  is  manifest  that  the  radius  of  tlie  arc  in  the  preceding 
formula  is  1,  or  unity,  which  may  easily,  from  the  principles 
of  homogeneity  in  the  members  of  the  equations,  be  reduced 
to  an  arc  whose  radius  is  r,  after  the  following  manner: 
Thus,  for  J/-  and  x^  in  the  first  two  equations,  write 

~   and     — 2-  and  they  become 

,               dy               ,      ,  dx 

dz  =  — ; — and    dz  = ; —  , 

which  are  easily  reduced  to 


,  rdy  ,      ,  rdx 

dz  =  — ; — - —     and     dz  — 


and  in  like  manner  the  remaining  equations  become 

,           r'dt  ,  7^1' 

dz  =  —7, -:  ,  dz= — 


,  T^ds  ,  T^ds' 

dz  =  — —  ,       dz=  — 


dz  = 


rdv  -  _  rdv' 

which  are  adapted  to  the  arc  z  whose  radius  is  r. 

Eemarks. — 1.  Diiferentials  that  are  not  of  the  preceding 
forms,  can  often  be  reduced  to  thena.     Thus 

^  :i 

d  Z 

is  equivalent  to 


/25  -  Ux' 


■m 


x" 


7^  REDUCTIONS  OF  FORMS. 

•which,  is  the  differential  of  a  circular  arc  whose  radius  is 

J  and  sin  =  a;  divided  by  5.     In  like  manner  the  differen- 

,  -dv 

tial  -s — i-r-r  is  reducible  to  — ; t:^ — r ;  which  is  the  dif- 


t^-S") 


ferential  of  a  circular  arc  whose  radius  =  1    and  tangent 
=  -  -y,  divided  by  db. 

2.  In  like  manner,  differentials  can  often  be  reduced  to 
those  of  known  logarithmic  forms.     Thus  the   differential 

-g — -^  is  reducible  to  the  known  logarithmic  differentials 

dx           —  dx  T      2adx 

H ,     and 


a  -\-  X        a  —  X  XT  —  a* 

.     ,     ,  ,                 dx             dx 
IS  equivalent  to , 

^  X  —  a        X  -\-  a 

which  are  differentials  of  well-known  logarithmic  forms. 

(10.)  We  will  conclude  this  section  by  noticing  some  of  the 
more  important  properties  of  the  expressions 

e^^-^  —  cosa?+  sin  a?  V~l  and  e~'^^~'^  =  cos;??— sina?  V  —  1, 
or  their  equivalents 

cos  X  = -„ and  sm  x  = tc-- ; 

2  Sv^TZTi 

see  page  53. 

It  is  manifest  that  for  the  fii-st  two  of  these  forms,  we  may 
take  e^'^^~'^  —  cos  a?  db  sin  a?  V  —  1 ;  by  using  the  upper 
signs  (in  the  ambiguous  signs)  for  the  first,  and  the  lower 
signs  for  the  second. 


DE  moivre's  formula.  75 

If  mx  is  put  for  x ,  we  sliall  have 

or,  because 


±  m  X  V  _  1  —  (fi  ±x  ^—  1  \  w 


(^±x  ^-ij'^z^  (cos  aj±  sin  a?  4^  —  1)"*, 


we  stall  get 

(cos  a?  db  sin  a?  I^^l)'"  =  cos  mx  ±  sin  ma?  4^—1  . . .  {f) ; 

wliicli  is  called  De  Moivre^s  Formulce. 

Expanding  tlie  first  member  of  this  equation  according  to 
the  ascending  powers  of  ±  sin  a?  -/  —  1  by  the  Binomial 
Theorem,  and  equating  the  real  and  imaginary  parts  of  the 
members  of  the  resulting  equation,  separately,  we  readily  get 

cos  mx  =  cos"*  X  —  — ^— — -  cos'"  ~  ^  x  sin^  x 
_j ^ ^v   _^.v /  cos"*-'*aJsm*aj— ,&c., 

JL  .  Z  .  O  .  4: 

and 

•       /      m    1         (?/?  — l)(m  — 2)       ^    o    .  , 
sm  maj  =  m  smaj  I  cos"*  ~^  a? ^r^ cos"*  ~  ^  sm'^  x 

,   {m  -  l)(m  --^  ^){m  -  3)(m  -  4)  ^    .    •  .  o    \ 

+  ^^ ^^^^ 7~r-i-^~^ X  cosaj"*-^  sm^a?— ,  &c. 

2. 3. 4. 0  '        / 

If  in  these  equations  we  successively  put  m  =  2^m  =  3, 
&c.,  we  get 

cos  2x  =  cos-  X  —  sin-  x  =  cos^  x  —  (1— cos^  a?)  =  2  cos^  a?  —1, 

sin  2a7  ==  2  sin  a?  cos  a?, 

cos  3a7  =  cos'^  a;— 3  cos  x  sin^  x  =  cos'^  a?  —  3  cos  a?  (1— cos^  a?) 

=  4  COS'^  37—3  COS  X, 

sin  3aj  =  3  sin  a?  cos^  x  —  sin''  a?  =  3  sin  a?  —  4  sin^a?, 
and  so  on. 


76  IMPORTANT  FORMULiE. 

If  in  the  expressions  for  cos  x  and  cos  ma?,  sin  x  and 

sin  inx^  we  put  o'^ ~'^  =  y,  and  of  course  e~''*^-~^  =  -,  then 

1  1 

we  get      2  cos  x  =  y  -\ — ^ ,     2  cos  mx  =  y""  +  -^.^ 

2  sin  X  V—  1  =  y ,      and     2  sin  mx  V—\  =  v"' -. 

Supposing  m  to  be   a  positive  integer;   by  raising  the 

members  of  2  cos  x  =^  y  +  -  io  the  mAh  power,  and  uniting 

the  first  and  last  terms,  the  second  and  last  but  one  terms, 
and  so  on ;  we  shall  evidently  have 

2»  cos"  a,  =  (y-  +  A)  +  m  (y"-=  +  -~) 

.    m  (m  —  1)  /  ^   .         1    \ 

If  in  is   an   odd  number,   since  y""  -\ — -  =  2  cos  7?ix, 

iT'"^  H ;;r:2  ~  ^  cos  {m  —  2)  a;,  and  so  on,  we  readily  get 

2        cos  mx  =  cos  mx  +  m  cos  (m  —  2)  a?  + 


cos  (tti  —  4),'»  +,  &c.,  until  the  number  of  terms  = 


1.2 

?yi  +  1 


2 
When  7?i  is  an   even  number,  we   have   2'"~*  cost's?  = 


,         _..  mim—X) 

cos  mx-{-m  cos  [m  —  2)  a?  +   — j— — ^  cos  (w— 4)  a?  +,  &c., 

-  terms  containing  cosines ;  to 

^  m(m-l)x...x(y  +1^ 


untn  there  are  —  terms  containing  cosines ;  to  which  must 


be  added  the  term 


1-2X xy 


IMPOKTANT  FORMULAE.  77 

If  in  these  formulae  we  put  1,  2,  3,  &c.,  successively,  for 
7??,  we  readily  get  the  following 

TABLE. 

1.  COS  a;  =  cosaj; 

2.  2  cos^  a?  =  cos  23?  + 1 ; 

3.  4  cos^  X  =  cos  3a? -f  3  cos  x ; 

4.  8  cos"*  X  =  cos  4;r +4  cos  2;c+3  ; 

5.  16  cos^  a?  =  cos  5a? +  5 cos 307 -1-10  cos  a?; 

6.  32  cos^  a?  =  cos  Ga?  +  6  cos  4a?  + 15  cos  2.r  4- 10 ; 

7.  64  cos^a?  =  cos  7a? +  7 cos 5a? +  21  cos3a?-h35  cos  x; 

8.  128  cos^  X  =  cos  8a? +  8  cos  6a? +28  cos  4a? +56  cos  2a?+35 ; 
and  so  on,  to  any  extent  that  may  be  desired. 

If  m  is  an  even  number,  and  the  members  of 

2  sin  a?  V~l  =^  y 

are  raised  to  the  mth  power,  then,  by  proceeding  as  before, 
we  shall  clearly  have 

±  2"*-^  sin*"  a?  =  cos  ma?  —  m  cos  (m  —  2)  a? 

.        7n  {771  —  1) 

H -:r-^ — -  COS  {m  —  4:)x  — ,  &c. ; 

1 .  Ji 

noticing,  that  +  must  be  used  for  ± ,  in  the  first  member  of 
the  equation,  when  m  is  exactly  divisible  by  4,  and  that  — 
must  be  used  when  it  is  divisible  by  2,  or  not  divisible  by  4. 

It  may  be  added,  that  there  will  here  be  -^  terms  containing 
cosines ;  together  with  the  term 


± 


^m{m.-  l)x....x  (y  + V 

1 .  2x X  -TT- 


78  IMPORTANT  FORMULJE. 

in  whicli  4-  must  be  used  for  ±  when  rn  is  divisible  by  4 ; 
and  wben  in  is  not  divisible  by  4,  we  must  use  — . 

When  m  is  an  odd  number,  by  proceeding  as  before,  we 
shall  have 

±  2"*~^  sin"*  X  =  sin  mx  —  m  sin  (m  —  2)  x 

H Y~2 —  ^^^  ^^^  —  4)  a?  — ,  &c., 

until  the  number  of  terms  equals  — ^ —  ;  noticing,  that  + 

must  be  used  for  ±  in  the  first  member  of  the  equation, 
when  m  —  1  is  divisible  by  4 ;  and  that  —  must  be  used  in 
the  contrary  case. 

If  1,  2,  3,  4,  5,  &;c.,  are  successively  put  for  in  in  the  pre- 
ceding formulae,  we  readily  get  the  following 

TABLE. 

1.  sin  X  =  sin  a? ; 

2.  —  2  sin^a?  =  cos2a?— 1; 

3.  —4  sin' a?  =  sin  3a?— 3  sin  x ; 

4.  8  sin*  X  =  cos  4,r — 4  cos  2a? + 3 ; 

5.  16  sin^a?  =  sin  5a;— 5  sin  3aj-f  lOsina;; 

6.  —32  sin''  x  =  cos  6x—6  cos  4a; +  15  cos  2a;— 10 ; 

7.  —64  sin^a;  =  sin  7a?— 7  sin  5a; +  21  sin  3a;— 35  sin  x; 

8.  128  sin^  a;  =  cos  8a;— 8  cos  6a? +28  cos  4a; +56  cos  2a? +  35; 
and  so  on,  to  any  required  extent. 

Resuming  the  simultaneous  equations  2  cos  x  =  y  -\ — , 

and     2  cos  mx  —  if  ^ — -^,  from  p.  76 ;  it  is  easy  to  per- 
ceive thjvt  they  are  equivalent  to  the  equations 
y^  —  2y  cos  a;  +  1  =  0,     and    'y^'^  —  2y'"  cos  ma;  +  1  =  0. 


FACTORS  OF  y"'"'  —  2y"'  COS  0  +  1  =  0.  79 

Because  these  equations  are  coexistent,  it  is  clear  that  the 
first  is  a  quadratic  factor  of  the  second. 

If  we  have  an  equation  of  the  form 

yim  _  2y«  COS  6>  +  1  =  y'^"'  —  22/"^  cos  {6  +  ^nz)  +  1  =  0, 
since  cos  6  =  cos  {0  +  2nn\  n  being  an  integer ;    then  we 


shall 


have  y^  —  22/  cos  I )  +  ^' 


for  the  general  representative  of  its  quadratic  factors.  Pat- 
ting successively,  0,  1,  2,  3,  &c.,  to  /i  =  m  —  1  for  n  in  the 
quadratic  factor,  we  clearly  get 

2^2^/1  _  2y^  cos  0  +  1  ==  \if  —  2?/  cos  —  +  l) 

X  {y'-  2y  cos  ^-^  +  l)  x  {y'  -  2y  cos  ^^—  +l),  &c., 

to  m  factors.  It  is  evident  that  these  factors  are  different 
from  each  other,  and  that  they  are  the  only  quadratic  fac- 
tors which  the  equation  can  have  ;  since  t^  ==  r/z,  ?i  =  m  +  1, 
n^=.m  ■\-%  kc,  will  merely  give  repetitions  of  the  factors 
found. 

Thus,  the  quadratic  factors  of 

since  cos  0  :=  ^    or   6  =  60°,  will  easily  be  found  to  be 

y'  -  1.8793852 .  y  -f  1,        y' -  1.5320888 .  y  +  1, 
and  2/^-0.3472964   y  +  1; 

and  in  the  same  way,  since 

gives  cos  0  =  —  ^ ,  we  readily  get  6?  ==  120°,  and  thence  we 


80  FACTORS  OF  y-'"  —  ly"^   COS  0  +  1  =  0. 

shall  have  f  -  1.5320888  .y  +  1,  y- -  0.3472964 -2^+1, 
and    y^  —  1.8793852 .  y  +  1,    for  the  quadratic  factors. 

If  we  have  an  equation  of  the  form  y'"*  —  2ay"*  -}- 1  =  0, 
in  which  a  is  numerically  not  greater  than  unity,  it  is  clear 
that  it  may  in  like  manner  be  resolved  into  quadratic  fac- 
tors. Consequently,  if  each  quadratic  factor  is  resolved  into 
its  two  simple  factors,  the  roots  of  the  proposed  equation 
will  be  known. 

If  a  =  1,  the  equation  becomes 

1      .                              .»      r.           2n7r 
having  ^  —  zy  cos h  1 

for  its  general  quadratic  factor,  since  cos  Inn  =  1.  Putting 
0,  1,  2,  3,  .  .  .  .  to  m  —  1,  inclusively  for  n,  the  particular 
quadratic  factors  will  be  found  to  be 

*      rt           47r       ,                 .       c,       rt           2(m— l)7r       . 
if  —  2y  cos h  1  .  .  .  .  to  v"  —  2y  cos  -^ — h  1, 

for  the  last  factor.     Because 

2(m  — l)7r       ^         27T 

__!^ _^_  =  27r , 

7/2/  m 

it  is  clear  that 

{2m  —  1)  27r 

cos  ^^ ^n  =  cos  — , 

7 a  m 

and,  in  like  manner, 

2  (m  —  2)  TT  47r 

cos  — ^ —  =  cos  — , 

m  m 

and  so  on  ;   consequently,  for 

„      ^  2(m  — l)7r      . 

j^  _  2y  cos  -^———-  +  1, 

.       ^           27r      , 
we  may  write  y-  —  2y  cos 1- 1  ; 


PACTOKS   OF    if"^  —  2?/""   COS   0  -f  1  =  0.  81 

for  y^  —  2y  cos  — ^ ^ h  1, 

we  may  write  y^  —  2y  cos  — — h  Ij 

and  so  on. 

Hence,  we  shall  have 

(^-  -  ly  =  (y  _  1)= .  (y^  -  22/cos  ?^  +  l)' 
X  {y'-  2y  cos  ^  +  1  )  ,  &c., 

to  — jz —  factors,  when  7n  is  an  even  number ;  and  to  — ^ — 

factors,  when  m  is  an  odd  number.     Consequently,  extract 
ing  the  square  roots  of  these  equal  products,  we  shall  have 

2/"'-l=(y-l).  (y^-2y  cos  ^  +  l)  .  (y»  -  22/cos  J  +  l) 

&c.,  to factors  when  m  is  even,  and  to factors 

when  7)1  is  an  odd  number. 

Thus  the  factors  of  y^—  1  =  0,  are 

y  —  1,  y'^—2y  cosy  +  1,  y'^  —  2y. cos  y  +  1,    and  y  +  1 ; 

and  those  of  2/^  —  1  =  0,  are 

2/  —  1 ,  y"  —  2y  cos  y  +1,     and      y^  —  2y  cos  y  +  1- 

In  like  manner,  if  a  =  —  1 ,  our  equation  becomes 
y2m^2y-4-l==(y"^  +  l)'=0; 
whose  general  quadratic  factor  is 

2    ,     _  2717T+7r 

y'+2y.cos ——  +  1, 

since  cos  i^n-n  +  tt)  =  —  1. 

4* 


82  FACTORS  OF   xf""  —  %f^  COS  <?  +  1  =  0. 

Putting  0,  1,  2,  3,  to  ??i  —  1  inclusive,  for  n^  and  tlien 
proceeding  as  before,  we  get 

f  -  2y  cos  ^  +  1 ,  2/'  -  2y  cos  ^  +  1, 

5^        -.     .      <.       n            2(7Ai  — l)7r  4- rr 
y'  —  2y  cos 1-1,  to  y  —  2y  cos  -^ f-  1. 

Because 

2{tn  —  i)n-\-n  n       2{7n  —  2)  rr -\- n _  2Tr 

m  7n  m  m 

and  so  on ;  the  factors  may  clearly  be  written  in  the  forms- 
(y-2!/cos^+  1  y  ,  (/  -2yoos  ^^  +  1  )', 


(y^-2ycos^  +  l)\ 


to  -r-  factors  when  m  is  an  even  number  and  to  — ^ —  fac- 

tore  when  m  is  an  odd  number. 

Ilence,  as  before,  the  factors  of  y"*  +  1  =  0,  are  expressed 

by  y^  —  2y  cos hi,  2/^  —  2y  cos  —  +  1,    and  so  on,   to 

or  — - —  factors ;  accordingly,  as  m  is  an  even  or  an  odd 

number. 

Thus,  the  factore  of  y^  +  1  =  0,  are 

f  -  2ycos  g-  +  1,  y'  4-  2y  cos  ^  +  1  =  y'^  +  1, 

and  2/'  —  2 y  cos  y  +  i  ; 

while  the  factora  of  y'  +  1  =  0 ,  are 
y^  —  2y  cos  7  -f-  1,  y^  —  2y  cos  -v-  +  1,    and    y  +  1. 


INTERESTING   PROPERTIES   OF   THE   CIRCLE. 


83 


It  results  from  what  lias  been  done,  that  any  equation  of 
the  form  a?"  ±  a"  =  0,  can  be  resolved  into  factors.  For  put 
a?"  =  a"y" ,  and  the  equation  is  readily  reduced  to  the  equiv- 
alent equations  y"  +  1  =  0  and  y"  —  1  ==  0 ,  whose  roots 
can  be  found  as  before. 

It  is  manifest  that  any  equation  of  the  form 
ar"*  —  2aaj"»  -\- h  =  0, 

can  be  reduced  by  the  rules  of  quadratics  to  equations  of  the 
preceding  forms,  and  their  roots  may  be  found,  as  before. 

It  may  here  be  proper  to  notice  some  interesting  proper- 
ties of  the  circle,  that  result  from  what  has  been  done 

c 


Thus,  let  AA'B,  &c.,  be  the  circumference  of  a  circle 
whose  center  is  O,  and  radius  K ;  then,  supposing  the  circum- 
ference is  divided  into  any  number  wi  of  equal  parts  AB,  BO, 
&c. ;  if  from  any  point  P,  in  the  plane  of  the  circle,  the 
straight  lines  PA,  PB,  PC,  &c.,  are  drawn  to  the  points  of 
division  of  the  circumference,  we  shall  have  the  equation 

OP-"*  —  20P'^  X  OA"^  cos  m  (AOP)  +  AO-"* 

=  2/'""  —  2^"*  cos  (9  +  1; 

where  we  represent  the  radius  E  =  AO  by  1,  or  unity,  PO 


84      DE  moivre's  and  cotes'  properties  noticed. 

by  y,  and  the  angle  POA  by—.    We  also  have  from  tbe 
triangles  POA,  POB,  POC,  &c., 

AF  =  2/=  -  2y  cos  ^  +  1, 

BF  =  f-2y  cos  POB  +  1  =  ?/*  -  2y  cos  ?i-—  +  1, 

m 

PC^  =  y=  -  2y  cos  ^—  +  1,  &c. ; 

consequently,   agreeably  to  De   Moivre's   Property  of  the 
ClrcUy  we  shall,  from  what  is  shown  at  p.  79,  have 
^m  _  2ym  cos  (9  +  1  =  PA-  X  PB^  X  PC'  X ,  &c., 

to  the  square  of  the  line  drawn  from  P  to  the  last  point  of 
division  of  the  circumference. 

If  the  angle  POA  =  0,  or  A  falls  on  OP,  the  preceding 
equation  becomes 

^m  _  2y-  4-  1  =  (y-_  1)2  =  PA2  X  PB=  X ,  &c., 
or  ±  (y'"  -  1)  =  P A  X  PB  X  PC  X ,  &c. 

If  the  arcs  AB,  BC,  &c.,  are  each  bisected  in  A',  B',  &c., 
then,  since  tlie  lines  drawn  from  P  to  all  the  points  of  divi- 
sion will  be  doubled  in  number,  the  preceding  equation  will 
become  (for  all  the  points  of  division  of  the  circumference), 
±  (/''^  -  1-'")  =  PA  X  PA'  X  PB  X  PB'  X ,  &c., 

=  ±  {y'""-  V^)   X  PA'  X  PB'  X ,  &c. ; 
which  gives 

^r>._im  =  2/"  +  l"N    or  y^'+l  =  PA' x PB' x PC x ,  &c.: 

noticing,  that  the  equations  ±  {y^^—l)  =  PA  x  PB  x  PC  x , 
&c.,  ^''^  -i- 1  =  PA'  X  PB'  X  PC  X ,  &c.,  are  called  Cotes's 
Properties  of  the  Circle;  see  pp.  32  and  33  of  Young's 
"  Differential  Calculus." 


SINGULAR  PROPERTIES   OF  THE   CIRCLE.  85 

."Remarks. — There  are  one  or  two  singular  properties  of 
circular  functions  that  it  may  not  be  improper  to  notice  in 
this  conuection. 

Thus,  resuming  the  equation  e^^~^  =  cos  x  +  sin  a?  l/— 1, 

from  p.  53,  and  putting  a?  =  -,  we  have 

e^'^'^'^V-l,     or    e~'^=:{V^)''~'; 
which,  expanded  according  to  the  ascending  powers  of  x,  by 
{b"\  given  at  p.  51,  gives 

for  one  of  the  properties. 

And  by  taking  the  hyperbolic  logarithms  of  the  members 

_7r  _  -| 

of     e   2"—  (|/_- !)*'-!,   we  have  —  -  =  V^l  x  ^  log  —  1, 

or  TT  =:  —  4/  —  1  log  —  1,  for  the  other  property :  noticing, 
that  TT  =  the  semicircumference  of  a  circle  whose  radius  =:  1, 
and  that  e  stands  for  the  base  of  hyperbolic  logarithms.  See 
pp.  33  and  U  of  Young's  "Differential  Calculus." 


SECTION  HL 

VANISHING    FRACTIONS. 


(1.)  When  tlie  numerator  and  denominator  of  a  fractional 
expression  are  each  reduced  to  naught  or  vanish,  bj  giving 
a  particular  value  to  a  common  variable,  the  expression  is 
called  a  vanishing  fraction. 

Thus,  —TT 2\  ^  ^  vanishing  fraction :  since,  by  putting 

a  (fl/  —  a  ) 

a  for  X,  it  is  reduced  to      .  ,,    ■  .,,  =  t:.     It  is  clear,  from 
'  a{a^—a^)       0 

aj"—  a^  (x  —  a)  (x^-\-  xa  +  a^) 


a  (x^—  a^)  a  {x  —  a)  {x -\- a) 

that  it  is  reduced  to  the  form  ^,  bj  putting  a  for  a? ;  since 

the  factor  x  —  a  (which  is  common  to  the  numerator  and 
denominator)  becomes  a  —  a  =  0. 

It  is  hence  evident,  that  vanishing  fractions  result  from 
the  vanishing  of  factors  that  are  common  to  their  numer- 
ators and  denominators. 

(2.)  Because  the  quotient  arising  from  any  division  is  man- 
ifestly independent  of  any  factors  that  are  common  to  the 
dividend  and  divisor,  it  is  clear  that  by  erasing  such  factors 
from  the  dividend  and  divisor  (or  dividing  them  by  their 
greatest  common  divisor)  before  the  particular  value  is  put 
for  the  variable,  and  then  putting  the  particular  value  in  the 
result,  we  shall  get  the  true  valua 


ILLUSTRATIONS.  87 

Tlius,  since       —,-. ^  =  ^^— — ^— — ,^ -—-^ 

a  {x^—  a"")  a  {x  —  a)  {x  +  a) 

is  reduced  to 7 —  by  erasing  tlie  factor  x—a  from 

a{x  -}-  a)       -^  ^. 

its  numerator  and  denominator ;  then,  by  putting  a  for  x  in 
- — 7 — - — r—  ,  we  get,  after  a  slight  reduction,  ^  for  the  true 
value  of  the  proposed  fraction,  when  a  is  put  for  x  in  it. 

(3.)^  If  for  generality,  we  use  ^~  to  stand  for  any  vanish- 

Jj   X 

ing  fractional  form,  which  becomes  -  when  a  is  put  for  x ; 

then,  if  A  denotes  the  true  value,  we  shall  have  -  =  A . 

¥x 
To  find  A,  we  may  clearly  put  .j^^t—  =  A,  or  ¥x=A  x  F'a? ; 

Jd  X 

then  to  eliminate  the  vanishing  factor,  when  it  has  neither  a 

negative  nor  fractional  exponent,  we  may  differentiate  the 

members  of  Fx  =  A  x  F'a?  on  the  supposition  of  the  con- 

d¥x 
stancy  of  A,  which  will  give  A  =  'jr?T-  ?  ^^^  if   the    right 

member  of  this  for  x=  a  is  reduced  to  7: ,  we  may  evidently, 

as  before,  put  A  =  -wn^y- ,  and  so  on,  until  a  fractional  form 

Ct't:   X 

will  finally  be  obtained,  in  which  both  the  numerator  and 
denominator  will  not  vanish  when  a  is  put  for  x ;  which  will 
clearly  be  the  true  value  of  the  proposed  fraction. 

x^  —  3.'»  +  2 


Thus,  to  find  the  true  value  of  the  fraction 
when  X  =  1]  which  reduces  it  to  the  form 


Sx'-  6x^  +  S' 
0 
0* 


88  ILLUSTRATIONS. 

Here,  Fa?,  F'a?,  and  a,  are  represented  by 

ajs  _  3aj  +  2,     Sa^  —  6u;»  +  3,    and     1 ; 
consequently,  from 

d  ix"  -  3a;  +  2)  =  (3£r  -  3)  dx 
and  d{Zx^  -  Qs?  +  ^)  =  (123?^  —  12aj)  dx, 

we  have       A  ==  — r^ — --^^  =  -    when    xz=  1, 
V2a^—12x        0 

Hence  we  have 

d^{a^-Sx+2)     _      d{3x'-S)     _  6a?   -    ^ 

c^  (3a;^  -  ear'  +  3    '^  d  (ISa,-^  -  12a!)  ~    36a!2  -  12  ' 
/»  -j 

which  becomes  ^ r-^  =    r  ,  when  1  is  put  for  a?,  which  is 

oO  —  1^  4: 

the  true  value  of  A,  that  of  the  proposed  fraction,  when  1  Ls 
put  for  x  in  it. 

Fa? 
(4.)  Still  using  ^^  to  represent  a  fractional  form  that  be- 

Jj  X 

comes  ^,  when  a  is  put  for  x ;  then,  the  vanishing  factor  that  is 

common  to  the  numerator  and  denon;inator,  whatever  may 
be  its  nature,  can  be  eliminated  from  the  fraction  after  the 
following  manner : 

Thus,  put  a  +  h  for  x  in  Fa?  and  F'a?,  and  expand  these 
functions  by  Taylor's  Theorem,  or  in  any  other  way,  accord- 
ing to  the  ascending  powers  of  h ;  and  they  (by  omitting  the 
vanishing  terms)  will  evidently  be  reduced  to  the  forms 
Ah"  -f  BA*  + ,  &c.,  A'A^' -f  B'A*'  + ,  &c.   Hence,  we  shall  have 

Fa?  _Y{a-{-h)  _  AA^  +BA^-f,&c. 
F^a!  ""  F\a  +  h)  ~  A!h^'  ^-  BVi^^  +,  &c. ' 

and  hence  it  is  clear  that  -r->  A"-''',  when  a  is  put  for  a?,  ex- 
presses the  value  of  the  proposed  fraction.     Thus,  if  a  is 


ILLUSTRATIONS.  89 

greater  tlian  a\  it  is  clear  that  the  value  of  the  iraction 
equals  0,  since  -^  ti"-'''  =  0,  when  A  =  0  ;  when  a  --  a\  the 

v^alue  of  the  fraction  is  -^, ,  since  a  —  a'  =  ^  reduces  A''-"' 
to   A*'  =  1 ;    and   when   a'   is    greater   than  «,   the   value 

A^  ~   A'A^ 


A  A 

^,  A'*-"'  =    -TTiT^n::^  =  infinity  when  A  =  0,  on  account 


of  the  infinitesimal  divisor  A'*'""  in 


A'A«'-« 

Hence,  a  fraction  whose  numerator  and  denominator  are 
reduced  to  naught  by  a  particular  value  (a)  of  the  variable, 
may  be  found  by  the  following 

RULE. 

1.  Divide  the  differential  or  differential  coefficient  of  the 
numerator,  by  the  differential  or  differential  coefficient  of  the 
denominator,  and  substitute  the  particular  value  of  the  vari- 
able in  the  result ;  then  if  the  numerator  and  denominator 
of  the  fraction  thus  obtained  are  not  both  reduced  to  naught, 
it  will  be  the  value  of  the  vanishing  fraction. 

If,  however,  the  numerator  and  denominator  of  this  frac- 
tion vanish ;  then  we  must  proceed  with  the  second  differen- 
tials or  differential  coefficients  of  the  numerator  and  denomi- 
nator in  the  same  way  as  before ;  and  so  on,  until  a  fraction 
is  obtained  whose  numerator  and  denominator  do  not  both 
vanish  for  the  particular  value  of  the  variable ;  which  will, 
of  course,  be  the  correct  value  of  the  vanishing  fraction. 

2.  If  in  the  preceding  process  any  differential  coefficient 
becomes  infinite,  for  the  particular  value  {a)  of  the  variable ; 
then,  we  must,  as  at  p.  88,  change  the  variable  into  «  +  /?,  in 
the  numerator  and  denominator  of  the  proposed  fraction,  and 


90  EXAMPLES. 

develop,  by  particular  processes,  the  numerator  and  denomi- 
nator into  the  forms  AA".+  BA*  +  ,  &c.,  and  A7i'''  +  B7/-''  +  , 
&c.,  arranged  according  to  the  ascending  powers  of  A ;  then, 
as  at  p.  88,  the  true  value  of  the  vanishing  fraction  will  be 

A  A 

"^  expressed  by  -r^  A*'-'*',  when  A  =  0  ;  which  equals  0,  -7-7 » ^^ 
^  A. 

infinity,  axxiordingly  as  a  —  «'  is  positive,  naught,  or  negative. 

Eemark. — Examples  that  do  not  fall  immediately  under 
this  rule  can  often  be  reduced  to  it,  and  thence  their  values 
found. 

EXAMPLES. 

QT*    ___    Q^nfl   /yj*    _l_    lyfl 

1.  The  value  of  y^ j^ = s ,  when  1  is  put  for  a?, 

.     f]^—a^  ,    -  ,         ,  Za^  —  lax—x"      , 

13    vg — T-^ ;   and  the  value  of  — — — ,  when  x  =  a, 

0  —  0'  ox  —  Oct 

"    "  "^'  __ 

2.  To  find  the  value  of -=== — ,  when  x  —  a. 

Vx^  -  a' 

Put  a  ~\-  h  for  x,  and  the  expression  is  immediately  reducible 


*°  ^Su^ltuT+Zy'  ^^'''^'  ^y  P''"'°S  ^  =  0,  gives  /I 
for  the  answer. 

Otherwisa     Kepresenting  the  sought  value   by  A,   w^e 
easily  get  a^  —  Sax  +  2ar  =  A^{x^  —  a^), 

which  gives  2x  —  a  =  A"  (ar  -\-  xa  +  x^\ 

by  erasing  the  factor  x  —  a  from  its  members ;  consequently, 

putting  a  for  a?,  the  answer  is  A  =— =^. 

r  3a 


3.  To  find     ,  — ,  when  1  is  put  for  x. 


EXAMPLES.  91 

Putting  1  4-  A  for  a?,  the  expression  reducesto 

(3A  +  2A^)^  _  U  +  2A^)^  _  /27A-^  +,  &c.\^  _ 
(3A  +  3A^  +  A^)*  ~  (3 A  +  3A^  +  A^)^  ~  V  ^^''  +,  &c./    ~" 

(3 A  ±,  &c.)S 
wliicii,  by  putting  A  =  0,  gives  naught  for  the  true  value 
of  the  proposed  fraction,  when  1  is  put  for  x. 

4.  To  find  the  value  of  -^ ^  -. ,  when  x  =  a. 

or  —  ar      a  —  X 

When  X  =  a^  the  dividend  and  divisor  are  evidently  un- 
limitedly  great,  instead  of  being  infinitesimals,  as  in  the  pre- 
ceding examples. 

Performing  the  division  before  putting  a  for  «,  we  get 
1       ^  _1_  _      1      . 
or  —  x^    '   a  —  X  ~  a  -\-  X  ^ 

consequently,  putting  a  for  x,  the  answer  is   —  . 

X  1 

5.  To  find  the  value  of  the   difference   it  —  , , 

X  —  1        log  X 

when  X  =  1',  the  logarithm  being  hyperbolic. 

Reducing  the  terms  of  the  proposed  expression  to  a  com- 
mon denominator,  skives  the  fraction        '^'    ,    ,— ;  which 

{x—  1)  log  X 

is  under  the  form  of  a  vanishing  fraction. 

Dividing  the  second  differential  coefficient  of  the  numera- 

1 

tor  of  this  fraction  by  that  of  its  denominator  gives  - — '- 

X  vy 
for  the  quotient ;  which,  by  putting  .1  for  a?,  gives  -^  for  the 
answer. 


92  EXAMPLES. 


n 


6.  To  find  the  value  of  the  product  (a?  —  1)  tan  ^  a?, 
when  1  is  put  for  x. 

When  a?  —  1  =  0,  tan  -  x  becomes  tan  -  =  infinity ;  con- 
sequently, one  of  the  factors  equals  0,  while  the  other  is 
infinite. 

Since  tan   ^  x= ,   the  product  becomes , 

2  nx  '  ^  nx^ 

cot^  cot^ 

which  is  a  vanishing  fraction ;  since  its  numerator  and  denom- 
inator both  vanish  when  x  =  l. 

Consequently,  dividing  the  differential  coefficient  of  the 
numerator  of  this  fraction  by  that  of  its  denominator,  we 

2  X  sm'*  ^  X 
get ,  which,  by  putting  1  for  a?,  since  sin  h  —  1> 

o 
gives for  the  answer ;  and  in  much  the  same  way,  the 

tan  -  X 
value  of ,  when  a?  =  0,  is  infinite. 


e^ 


7.  To  find  the  value  of -. ,  when  x  =  0. 

a?  —  sm  a? 


From  (h")  page  51,  we  have 

Cu  sm  X 

e'  =  l-\-x  -^  — -r  +,&c.,and  <?'^"*=l4-sina?-h  T~ir  +>  ^o-  » 

consequently, 

(,x  _  ^sin.>^  -^  (a,  ^  sin  a>)  =  1  +  ^t.^-^  +,  &c., 

which,  by  putting  a;  =  0,  gives  1  for  the  answer. 

o    mu        1  ^  2a;  —  sin  a?         ,     af  —  a?       , 

8.  The  values  of ■ and    -^ ,  when  x  =  0 

X  1  —  x 

and  1,  are  1  and  0. 


EXAMPLES. 


9.  To  nnd  the  value  of -. r ,   when  a  is 

w*  —  a^ 

put  for  X. 

Put  a  +  A    for  X,   and   the  answer  will   be    found  to 

.     Ida  ^  1 

be  -J-  ,  or  ba  more  nearly. 

10.  The  values  of :: —    and    — — -. r ,  when  1 

x—1  ax~^  —  a~^ 

3 

and  a  are  put  for  a?,  are  1  and  — 

11.  The  value  of  —. ^ — r. r ,  when  a?  =  a,  is 

a^  —  2a^x  +  2ax^  —  a?^ '  ' 

unlimitedlj  great ;  and  that  of 

when  a?  =  a,  is  {^ay, 

12.  The  values  of 

ajn+i_fl^"  +  i  a?  —  bax  ■\- 4:0? 

and 


a;"  —  cC"  ^x"  —  lax  +  4a«  ' 

71  +  1 

when  «  is  put  for  x,  are a,  and  3. 

13.  The  value  of ^-- ,  when  cc  =  0,  is  ^  . 

For  most  of  the  preceding  examples  the  reader  may  be 
referred  to  pages  60  and  61  of  Young's  "  Differential  Cal- 
culus." 


SECTION  IV. 


MAXIMA  AND  MINIMA. 


(1.)  A  VALUE  of  a  function  greater  than  the  immediately 
preceding  and  following  values  is  called  a  maximum^  while 
a  value  less  than  those  values  is  called  a  minimum. 

Thus,  since  three  successive  values  of  a  function  of  any 
variable,  as  x,  may  clearly  be  expressed  by  the  forms 
¥  (x  —  h),  Yx,  and  ¥  {x  +  h);  ¥x  will  be  a  maximum  or 
mininum,  accordingly  as  it  is  greater  or  less  than  each  of 
the  other  values,  from  any  finite  value  of  h  (however  small), 
to  A  =  0. 

(2.)  Hence,  supposing  the  functions  ¥{x  —  A)  and  ¥{x-\-  h) 
to  be  converted  into  series  arranged  according  to  the  ascend- 
ing powers  of  A,  they  may  clearly  be  expressed  by  the  forms 

¥x  +  A  {- hy  -\-  B  {- hf  +,  &c., 

and  Faj  +  A(A)«4-B(Af +,  &c., 

in  which  A,  B,  &c.,  are  supposed  to  be  independent  of  A, 
while  the  index  a  is  less  than  h,  h  less  than  c,  and  so  on ; 
these  series  (like  the  functions  they  represent)  being  each 
less  or  greater  than  ¥x  from  a  very  small  value  of  A  to 
A  =  0.  It  is  clear  that  these  expansions  may  be  written  in 
the  forms 

F(aj-A)  =  F^+(-A)''[A  +  B(-A)*-«+C(-A)''-«  +  ,&c.], 
and   F  {x  +  A)  =  ¥x  +  h'  [A  +  BA''-'^  +  Ch'-''  +, &c.]  ; 
in  which  the  indices  h  —  a^c  —  a^  kc,  are  clearly  all  positive. 


DEDUCTION   OF  FORMULAS.  95 

If  A  is  different  from  0,  it  is  clear  tliat  so  small  a  finite 
value  may  be  given  to  A,  that  A  shall  be  greater  than  the  sum 
of  all  the  other  terms  within  the  braces,  in  the  expansions ; 
consequently,  when  F.»  is  a  maximum  or  minimum,  the 
terms  A(  —  A)"  and  AA""  must  accordingly,  each  be  negative 
or  positive.  Hence,  a  must  evidently  be  either  an  even  in- 
teger, or  a  vulgar  fraction  which  (in  its  lowest  terms)  has 
an  eveii  integer  for  numerator  and  an  odd  integer  for  its  de- 
nominator; and  A  must  be  negative  or  positive,  accordingly 
as  '¥x  is  a  maximum  or  minimum. 

(3.)  Regarding  x  and  h  as  indeterminates,  we  may,  by 
Taylor's  Theorem  for  the  above  formulas,  write 

,,      ^        d{^x)j       dX¥x)h''      d^(Fx)    A' 
and 

To  reduce  these  expansions  to  the  preceding  conditions,  we 
must  put  the  coefficient  of  h  equal  to  naught,  or  assume 

the  equation     '     ^  =  0;   and  the  expansions   will  be  re- 
duced  to  F(.  -  A  =  F.  +  -^-^  --  -  -~^J  j-^^^  +,  &c., 

and        F(..  +  h)  ^  F^  +  ^-^  S  +  ^^^'^^A  +'  ^^^ 

which   are  clearly   of  the  requisite  forms,   since  h?  is  the 
lowest  power  of  h^  in  them. 

When  a  function  is  a  maximum  or  minimum,  any  con- 
stant factor  or  divisor  of  it  may  be  omitted,  and  vice  versa. 
Also,  any  positive  power  or  root  of  a  igiaximum  or  mini- 
mum, must  also  be  a  maximum  or  minimum.     And  the  re- 


96  RULE   FOR  A  MAXIMUM   OR   MINIMUM. 

ciprocal  of  a  maximum  is  a  minimum  ;  and  that  of  a  mini- 
mum is  a  maximum. 

(4.)  It  is  manifest  that  the  maxima  and  minima  of  a  func- 
tion of  a  single  variable  may  be  found  by  the  following 

RULE. 

1.  To  find  when  y^  a  function  of  a?,  is  a  maximum  or 

minimum ;  put  the  first  differential  coefficient  ~~  =  0^  and 

find  the  real  roots  of  the  equation.     Substitute  each  real 

root  in  ~,  y|,  &c.,  until  the  first  which  does  not  vanish  is 

obtained;  then,  if  it  is  of  an  odd  degree,  it  can  not  corre- 
spond to  a  maximum  or  minimum  of  y ;  while  if  it  is  of  an 
even  degree,  it  will  correspond  to  a  maximum  or  minimum 
of  y,  accordingly  as  its  sign  is  negative  or  positive. 

2.  To  find  other  maxima,  and  maxima  that  may  result 
from  the  unlimited  increase  of  -^,  we  put  -j-  =  infinity; 

or,   which  comes  to  the  same,   we   assume  its  reciprocal 

dx 

-J-  =  0 ;  and  find  the  real  roots  of  this  equation.     Then,  the 

roots  which,  put  for  x  in  ?/,  make  it  greater  than  its  adjacent 
values,  will  give  maxima ;  while  those  which  make  y  less 
than  its  adjacent  values,  give  minima:  noticing,  that  those 
roots  which  do  not  make  y  a  maximum  or  minimum,  can  not 
correspond  to  the  maxima  and  minima  of  the  question. 

3.  If  any  real  root  of  -^  =  0,  w^ien  substituted  as  di- 

dx 

rected  in  1,  makes  jhe  first  differential  coefficient,  which  does 
not  vanish,  infinite,  then  the  true  value  of  the  term  must 


EXAMPLES.  97 

be  found  bj  tbe  ordinary  processes  of  algebra,  and  thence  the 
corresponding  maximum  or  minimum  may  be  determined. 

EXAMPLES. 

1.  To  find  tbe  maximum  and  minimum  of 

yz=:z^^—^x''+  l^x  —  7. 

Here  -#  =  0  becomes  ar*  —  3a?  +  2  =  0 ;  whose  roots  are 
ax 

(Px 
x  =  l  and  x  =  2.     Substituting  a?  =  1  in  -7^  =  2aj  —  3,  it  be- 
comes —  1,  which  being  negative  shows  that  if  we  put  1  for 
X  in  y,  we  shall  get  its  maximum.     Also,  putting  2  for  x  in 

-^  =  2x  —  3,  it  becomes  —  =  1 ;  which,   being  positive, 

shows  if  we  put  2  for  x  in  y,  we  shall  get  its  minimum 
value. 

2.  To  find  the  minimum  value  of  y  =  a^  —  {a  +  h)x  +  ab. 

Here  -f-  =  ^  becomes  2x  —  {a-\-h)  —  0,  which  gives 
X  =  — ^ — ,  and  -T3  =  2 ;  consequently,  putting  — -—  for  x 

m  2/,  we  have  y—  —  I  — « — )  ?  a  mmimum. 

3.  The  minima  values  of 

y  —  g?  —  2ax  -\-  o?  -\- 1  =  {x  —  af  +  5,  and  y  =  {x  —  «)*, 
are  evidently  y  =  h,  y  =  0 ;   while  y  =  (x  —  of,  admits  of 
neither  a  maximum  nor  minimum. 

4.  To  divide  2rfi  into  two  parts,  whose  product  shall  be  a 
maximum. 

Because  m-\-x  and  7n  —  x  when  added  equal  2m,  they 
may  clearly  stand  for  the  parts ;  consequently,  the  product 
of  the  parts  is  expressed  by   {m  +  x)  {m  —  x)  —  m^  —  ^^ 


98  EXAMPLES. 

which  is  clearly  a  maximum  when  x  =  0,  which  shows  that 
the  parts  are  equal 

Remark. — It  is  hence  easy  to  perceive  that  the  number 
nffi,  when  divided  into  n  equal  parts,  gives  m"  for  their 
maximum  product 

5.  To  find  the  maxima  and  minima  of  y  =  a  ±  (x  —  lif. 

Here,  we  have  -^=  ±^  («»  —  ^)~^  =  ±  ^ r ;  which 

dx  3^         ^  ^{x-bf 

shows  that  x  =  h  makes  —■  unlimitedlj  great,   or  reduces 
~-  =  -  {x—hy  to  naught,  agreeably  to  2  of  the  rule. 

By  putting  x^h  —  h,  we  evidently  have  y  =  a  ±  A* ; 
which  by  (2.),  p.  94,  makes  y  =  aja.  maximum  when  —  is 
used  for  ± ,  and  the  reverse. 

6.  To  find  the  maximum  and  minimum  of  y  =  a  ±  {x—Vf. 

get,  by  putting  -^  =  0,  aj  =  J,  which  makes  ~  =  infinity. 

Hence,  by  3  of  the  rule,  put  x  =  h  -{•  h,  and  we  get 
y  =z  a  db  (A)' ;  which  shows  that  by  using  —  for  ±,  x  =:  b 
makes  y  =  a,  a  maximum,  and  the  reverse. 

7.  Given  y  =  i^a^x  —  ax^  to  find  when  y  is  a  maximum 
or  minimum. 

i/ 
Here  we  easily  get  -  =  a-x  —  x"^  for  which  we  may  evi- 
dently take  u  =  (]^x  —  x" ,  and,  agreeably  to  the  remarks  at 
the  bottom  of  p.  95,  find  the  maximum  and  minimum  of  u. 

From    -^   =  0,     and    -7-5  =  —  6a7,  we  get  x  =  — -r   and 
fix  (M  '       o  ^3 


EXAMPLES.  99 

X  =  — -— ., ;  and  by  putttnor  _^  for  x  in  -7-^,  we  have  a 
negative  result;    wliicli  sKows   that    x  =  —^  makes   y  a 

yo 

maximum :  noticing,  that  x= —  makes  y  imaginary. 

y  o 

8.  To  solve  the  equations  x  -\-  y  +  2  =  a,  x"  -\-  y^  =  P, 
and  xy-  =  a  maximum  or  minimum  ;  or  to  find  x,  y,  and  2, 
from  the  equations  and  the  maximum  or  minimum  con- 
dition. 

Putting,  according  to  the  second  and  third  conditions, 
their  differentials  equal  to  naught,  we  evidently  have 

xdx  +  ydy  =  0,      and     2xydy  +  y^dx  =  0 ; 

consequently,  since  the  first  of  these  gives  ydy  =  —  xdx^ 
the  second,  by  substitution,  becomes  (y^  —  2,^)  dx  =  0,  or 
2/2  =  2z^ 

Hence,  the  second  of  the  proposed  equations,  by  putting 
2x^  for  y-,  is  reduced  to   Sx^  =  ¥;    whose  solution  gives 

a?  =  -—    and    x  — ~\ 

noticino^,  that  x  =  — :,  makes  xy^  a  maximum,  and  x  =  — — 

makes  it  a  minimum. 

Having  found  x^  we  easily  get  y  from  a^  -}-  y'^=  J-,  and 
thence  3  will  be  found  from  x  -\-  y  -\-  z  =  a. 

Kemarks. — 1.  It  is  hence  easy  to  perceive  that  we  may 
proceed  in  much  the  same  way  to  solve  all  questions  of  an 
analogous  nature. 

2.  The  preceding  solution  may  be  modified  as  follows: 
From  the  second  equation  we  have  y'^  =  h'^—a^,  which  reduces 
the  maximum  or  minimum  condition  to  the  maximum  or 
minimum  of  ¥x  —  x^ ;  consequently,  representing  this  by  w, 


100  EXAMPLES. 

we  have  to  find  x  suclx  that  u  =  h^x  —  Tf  shall  be  a  maxi- 
mum or  minimum. 

Hence,  from  y-  =  0,  or  J^  —  Sar'  =  0,   and  -r-^  =  —  6.7?, 

we  get  the  same  results  as  before. 

9.  Given      x  -\-y  -\-  z=  a^     and    a?"*  y"  s^, 
or  m  log  X  +  n  log  y  +  p  log  s, 

a  maximum,  to  find  a?,  y,  and  z. 

By  taking  the  difierentials,  we  have  dx  -\-  dy  -{-  dz  =  0, 

-,              7          7            1     ^c?a?      Tidi/       pdz 
or      dz  =■  —  ax  —  dy.     and    H =^  4-  - —  =  0  ; 

^'  X     ^    y    ^     z 

consequently,  substituting  the  value  of  dz^  we  have 

mdx       j)dx      ^  ndy       pdy  ^ 
X  z  y  z    "* 

which,  on  account  of  the  arbitrariness  of  dx  and  dy^  is  clearly 
equivalent  to  the  equations 

m       p       ^  m       X  ^     n        y 
=  0,     or    —  =  -      and     -  =  -. 

X        z  p        z  pi      z 

Hence,  to  the  sum  of  these  equations  adding  the  identical 
equation  -  =  -,     we  have 

m-\-n-{-v'x-\-y-\^z       a  ap 

^  =L =:  -     or     z  = 

p  z  z  7n  -\-  n  -\-  p 

TltZ  71 Z 

and  thence  from    x  =  —      and    y  =■  — ,   we  readily  get 

am  T  an 

X  = and    y  = 

m  +  n  -^  p  *         rn  -\-  n  -\-  p 

To  perceive  that  the  preceding  results  satisfy  the  required 
conditions,  the  reader  may  consult  Lacroix,  "  Calcul  Dif ." 
vol.  I,  p.  380. 


<1      n    ^  ^-   - 

^        '  f)  EXAMPLES.  101 

10.  To  find  a?,  sucti  tliat  -^ s  shall  be  a  maximum. 

'  X-  -{■  G^ 

According  to  what  is  stated  at  p.  95,  the  question  will 
be  solved   by  making  the  reciprocal  of  the   proposed   ex- 

pression,    or =  a?  H —  ,  a  minimum. 

Because  a?  x  —  =^  <?.   it  is  clear  that  the  minimum  of 

X 

X  -\ is  found  by  putting  x  z=z  c\  which  gives  x\c\\c\-\ 

X  '^ 

consequently  [see  my  Algebra  (24),  at  the  top  of  p.  197], 

we  must  have  x  A —  ereater  than  <?  +  <?  =  2  <?,  if  a?  and  —  are 

'  a?  ^  X 

unequal. 

11.  Given  the  angle  A  and  the  position  of  the  point  P 
between  the  lines  that  form  it,  to  draw  the  right  line  DE 
through  P  such  that  the  triangle  ADE  shall  be  a  minimum. 


Through  P  draw  right  lines  parallel  to  the  lines  that  form 
the  given  angle  and  meeting  them  in  B  and  C ;  then,  DPE 
being  drawn  to  cut  off  the  minimum  triangle,  the  triangles 
PBD  and  PCE  are  evidently  equiangular  and  of  course  sim- 
ilar, from  well-known  principles  of  geometry ;  and  the  area 
of  the  parallelogram  PBAC  is  evidently  given,  from  the 
data  of  the  question, 

Eepresenting  PB  =  AC,  PC  =  AB,  BD,  and  CE,  by  the 
letters  a,  J,  a?,  and  y,  we  get  from  the  similar  triangles  the  pro- 


102  EXAMPLES. 

portion  x  :  a  ::   h  :  y  =  —  ;  consequently, 

AD  =  a?  +  5  and  AE  =a  -\-y=a-\ =  _LJL_^ 

and  thence  we  liave  tlie  area  of  the  triangle  ADE  expressed 

by 

AD  X  AE  X  sin  A      a  (»+i)'        a  I        „,*%.. 

as  is  evident  from  the  well-known  expression  for  the  area  of 
a  triangle  in  terms  of  any  two  of  its  sides  and  their  included 
angle.  'tr 

From  the  preceding  expression,  since  sin  A,  -  ,  and  h  are 

given,  it  clearly  follows  (from  principles  heretofore  given), 

that  the  trianojle  will  be  a  minimum  when  x  -{ —  is  a  mini- 
°  X 

mum. 

Hence,  from  the  solution  of  the  preceding  example,  we 
must  have  a?  =  Z>,  or  BD  =  B  A,  from  which  it  clearly  follows 
that  making  BD  =  BA  =  J,  and  drawing  DPE,  DBA  will 
be  the  required  triangle ;  and  P  i)isects  DE. 

12.  "Oiven  the  sum  of  the  base  and  curve  surfice  of  a 
right  cylinder,  to  find  when  its  solidity  is  a  maximum." 

Let  r  and  A  stand  for  the  radius  of  the  base  and  height  of 
the  cylinder,  and  tt  =  3.14159,  &c.  ==  the  semicircumference 
of  a  circle  whose  radius  =  1 ;  then,  if  A  stands  for  the  sum 
of  tlie  base  and  curve  surface,  we  shall,  from  the  known  prin- 
ciples of  mensuration,  get  2?'-A  +  /'^rr  =  A  and  t-tJi  =  s  = 
the  solidity  of  the  cylinder  =  a  maximum.  From  these  con- 
ditions, we  readily  get  25  =  Ar  —  rV  —  a  maximum,  which 

gives    -7—  =  0  or  A  —  dm  =0   or  r  —  y  --  . 


EXAMPLES. 


103 


From  tlie  addition  of  2/'7rA  +  rV  —  A  and  A  —  3rV  =  0, 
we  get  h  —  r,  or  the  height  of  the  cylinder  equals  the  radius 
of  its  base. 

Remark. — In  much  tlie  same  way,  if  the  whole  surface  is 
given,  when  the  cylinder  is  a  maximum  we   shall    have 

r  =  y  -^  ,  and  h  —  2/",  by  using  A  to  represent  the  whole 

surface. 

13.  Find  the  longest  straight  pole  that  can  be  put  up  a 
chimney,  when  the  height  from  the  floor  to  the  mantel  =  a, 
and  the  depth  from  front  to  back  =  h 


Let  D  represent  the  mantel,  and  AB  the  pole  passing 
through  it,  meeting  the  floor  in  A,  and  the  back  of  the  chim- 
ney in  B ;  then  BE  =  a    and     DG  =  EF  =  h. 

Representing  AE  by  x,  the  right-angled  triangle  ADE 
gives  AD  =  |/  (a^  +  a?-),  and  then  from  the  similar  triangles 
ADE  and  ABF  we  have  the  proportion 

AE  :  AD  ::  AF  :  AB  =.  4^.  AF=  ^^'''  +  '^'^  {h  +  x)  = 

AE  X         ^  ^  ^ 

the  length  of  the  pole  =  a  maximum;  consequently, 
(a-  -\-  ar)l-  +  1 1  must  be  a  maximum.  Putting  the  difier- 
ential  of  this  equal  to  naught,  we  readily  get  the  equation 

X  -\-h ;,  (a^  +  «")  =  0, 

X-  ^  ^ 

which  gives  x  =  Vcvl) ,  as  required. 


104 


EXAMPLES. 


Otherwise.  Supposing  AB  to  be  tlie  position  of  tlie  rod, 
let  it  be  slightly  changed  into  the  position  A'B',  by  revolv- 
ing about  D ;  then  (ultimately),  its  change  A'C  at  the  ex- 
tremity A  must  equal  its  change  at  the  extremity  B,  and 
have  a  contrary  sign ;  consequently,  the  approximate  position 
of  the  rod  can  be  easily  found  by  trial. 

It  clearly  follows  from  what  has  been  done,  that  we  shall 

have  AD  :  DB  : :  tan  ang  A  :  tan  ang  B,or  x  :  h  :  ~  :  — ,  or 
—  =  —  ,  which  gives  x  =  \r^  the  same  as  before. 

Cb  X 

14.  To  find  when  the  cylinder  DIGrF  inscribed  in  the  cone 
ABC  is  a  maximum. 


Eepresent  the  base  and  height  of  the  cone  by  A  and  «, 

and  the  height  of  the  cylinder  by  a?,  then  a  —x  represents 

the  height  AE  of  the  cone  whose  base  is  DF  the  upper  base 

of  the  cylinder.     From  well-known  principles  of  geometry, 

we  have 

AH^  :  AE^  ::  baseBC  :  baseDF  = 

j-TfT,  X  base  BC  =  -^  x  (a  —  xf: 

consequently,  multiplying  this  by  x,  the  height  of  the  cylin- 

der,  we  have  — j-  («  —  a?)^  a?  lor  its  contents. 


EXAMPLES.  105 

Hence,  because  —r  and  a  are  invariable  (a  —  xf  x  must  be 

a  maximum,  whose  differential,  put  equal  to  naugbt,  gives 
—  2xdx  (a  —  x)  -\-  (a  —  xy  dx  =  0  or  —  2x  -^  a  —  x=^  0. 

This  solved,  gives  x=  ^  ,ov  the  height  of  the  cylinder  is 
o 

one-third'  of  that  of  the  cone. 

Remark. — It  may  be  shown,  in  much  the  same  way,  that 
the  height  of  the  maximum  rectangle  in  any  triangle  is  half 
the  height  of  the  triangle. 

15.  "  To  cut  the  greatest  parabola  from  a  given  cone." 


Let  ABC  be  a  triangular  section  of  the  cone  by  a  plane 
passing  through  its  axis  at  right  angles  to  its  base,  and  sup- 
pose that  the  sought  parabola  passes  through  F  in  BC,  then, 
drawing  the  lines  GE  and  FD  through  F,  parallel  to  the 
tangent  to  the  circumference  of  the  base  at  0  and  to  the 
side  of  the  cone  AC,  meeting  the  circumference  of  the  base 
in  the  points  Gr  and  E  and  the  side  AB  of  the  cone  in  D, 
the  curvilinear  section  GDE  of  the  conical  surface  and  the 


106  EXAMPLES. 

plane  of  the  lines  GE  and  FD  will,  according  to  the  com- 
mon definition,  be  a  parabola ;  having  DF  for  its  axis  and 
D  for  its  principal  vertex,  FE  and  FGr,  which  are  evidently 
equal  and  perpendicular  to  BC,  being  called  ordinates  to  the 
axis.  If  through  D,  DH  is  drawn  parallel  to  FC,  and  DI 
drawn  above  DII  so  as  to  make  the  angle  IIDI  equal  to 
the  angle  A  or  FDB  ;  then  HI,  the  part  of  the  side  of  the 
cone  between  the  lines  HD  and  ID,  will  be  what  is  called 
the  principal  parameter  or  latus  rectum,  of  the  parabola,  it 
being  the  parameter  or  latus  rectum  of  its  axis.  Since 
the  angles  D  and  H  of  the  triangle  HDI  are  equal  to  the 
angles  D  and  F  of  the  triangle  FDB,  these  triangles  are 
clearly  equiangular  and  give  the  proportion  HI  :  DH  or 
FC::FB  :  FD,  or  its  equivalent,  HIxDF=CF  xFB4fE^ 
by  a  well-known  property  of  the  circle.  Kepresenting  HI 
by  ^,  DF  by  a?,  and  FE  by  ?/,  the  preceding  equation  be- 
comes px  =  y^,  the  well'known  equation  of  the  parabohi ; 
which,  by  knowing  p  and  assuming  a?,  will  enable  us  to 
find  the  corresponding  values,  +  y  and  —  ?/,  of  y,  so  that 
the  curve  may  clearly  be  constructed  by  points,  according  to 
the  common  methods.     Because  the  area  of  the  parabola 

2  4 

GDE  equals  -  G-E  x  DF  =  -  xy^  it  is  clear,  since  the  area  is 

a  maximum,  that  xy  must  also  be  a  maximum.  If  we  rep- 
resent the  diameter  of  the  base  BC  by  a  and  BF  by  2,  we 
shall  get  CF  =  a  —  s ;  which  give  2/'  —  az  —  s-,  from  a  well- 
known  property  of  the  circle.  Because  the  angles  of  the 
triangle  BDF  do  not  change  for  different  positions  of  the 
parabola,  it  is  clear  tliat  DF  will  vary  as  BF  or  .s ;  conse- 
quently, xy  may  be  represented  by  z  \/{az  —  z-)  and  a/^  —  z^ 
must  be  a  maximum.     By  putting  the  diiferential  of  this 

equal  to  naught,  we  have  Baz^  ■—  42^  =  0,  which  gives  z  =  —j 


EXAMPLES. 


107 


wbicli,  of  course,  gives  the  position  of  ttie  parabolic  section, 
wlien  it  is  a  maximum. 

16.  "To  find  the  position  of  a  straight  rod  or  beam,  when 
it  rests  in  equilibrio  on  a  prop  in  a  vertical  plane,  having  one 
of  its  ends  in  contact  with  a  vertical  wall,  which  is  at  right 
angles  to  the  vertical  plane  of  the  rod." 


Let  BC  be  half  the  beam  (supposed  of  uniform  density 
and  dimensions)  on  the  prop  PR,  and  having  its  end  B  in 
contact  with  the  vertical  wall  EB\  whose  plane  cuts  the  ver- 
•tical  plane  of  the  rod  perpendicularly ;  then,  through  P 
draw  DE  perpendicular  to  the  plane  of  the  wall,  and  DO 
through  G,  the  center  of  gravity  of  the  beam,  perpendicular 
to  the  direction  of  EP,  meeting  its  production  in  D ;  then, 
since  the  beam  is  in  equilibrio,  it  results  from  well-known 
'principles  of  mechanics  that  DO  must  be  a  maximum.  Put 
BO  =  half  the  length  of  the  beam  =  Z>,  and  PE  the  dis- 
tance of  the  prop  from  the  wall  =  a,  and  represent  the  angle 
BPE  =  CPD  by  (^ ;  and  we  shall  have 

BO  sin  </>  =  2>  sin  0  =  BE  +  CD, 
also  we  have  BE  =  PE  x  tan  <^  —  a  tan  (/>, 

and  hence,  by  subtraction,  we  have 

Z*  sin  </)  —  a  tan  </>  =  DC  =  a  maximum. 

Hence,  putting  the  differential  of  this  equal  to  naught,  we 


108 


EXAMPLES. 


have  h  cos  ^ 
required. 


3   /(L 

a  —-  QO^  ^  =  0,  whicli  gives  cos  «/>  =  y  ^, 


as 


Another  solution. — ^Let  the  figure  be  constructed,  and  the 
same  notation  used  as  before :  then  let  the  beam  be  slightly 
changed  in  position,  first,  by  giving  it  a  slight  angular  motion, 
in  its  vertical  plane,  about  P  until  it  takes  the  position  C'Gr ; 
second,  by  sliding  it  along  P,  without  any  angular  motion, 
until  G  coincides  with  B'  in  the  vertical  wall. 

From  P  as  a  center,  with  radii  PC  and  PB  describe  the 
arcs  CC^  and  BG,  cutting  BO  and  B'C  in  the  points  C  and  Q>\ 
B  and  G,  and  suppose  the  horizontal  line  through  C  meets  a 
vertical  line  through  C  in  I ;  then  IC  clearly  represents  the 
vertical  motion  of  the  point  C  (or  the  center  of  gravity  of 
the  beam),  resulting  from  the  angular  motion. 

Because  the  arcs  QG'  and  BG,  on  account  of  their  (sup- 
posed) minuteness,  may  be  regarded  as  right  lines,  which  cut 
BC  and  B'C  perpendicularly  on  account  of  the  minuteness 
of  the  angular  motion  cZ</),  it  is  easy  to  perceive  that  the 
triangles  CIC  and  CDP  will  (ultimately)  be  similar. 

It  is  also  easy  to  perceive,  if  the  perpendicular  to  the  wall 
through  G  meets  it  in  H,  that  the  triangles  BPE  and  GBH 
will  (ultimately)  be  similar  to  each  other,  and  to  the  triangles 


EXAMPLES.  109 

DPC  and  C'lC ;  hence,  we  have  h  x  d<p  =  CC  +  BG 
and  CI  =  CC^  cos  <^,  BH  =  BGr  cos  </>,  and  of  course 
CI  -{- BR  =  h  cos  (pdcp. 

If  B'H  =  CI,  it  is  easy  to  perceive  tliat  the  center  of 
gravity  of  the  beam  will  be  raised  by  sliding  it  along  P  (or 
keeping  its  end  in  contact  with  the  vertical  wall),  through 
CI ;  consequently,  since  the  center  of  gravity  neither  ascends 
nor  descends,  the  beam  must  clearly  be  in  equilibrio,  as 
required.     Now  from  the  triangle  BPE,  we  have 

BP  =a  X  sec  (b  = , . 

cos  </) 

and  thence  from  BPG  we  get  BGr  = ;  consequently, 

since  ang  BG-B^  differs  insensibly  from  a  right  angle,  we 

have  BG  sec  <;^  =  -^  =  BB'. 

cos-0 

Hence,  when  the  beam  is  in  equilibrio,  we  must  from 

BB'  =  CI  +  BH  =  h  cos  (bc/fp  have  h  cos  (hdcb  =  — ~- :  or, 

cosrcp 

from  a  slight  reduction  cos  (f)  =  y  j-  ^   the   same  result  as 

found  from  the  preceding  solution. 

Eemark. — Thus,  it  clearly  follows  that  the  question  admits 
of  a  most  elegant  solution,  which  does  not  require  the  use  of 
any  principles  that  depend  on  maxima  or  minima. 

17.  "Given  two  elastic  bodies  A  and  0,  to  find  an  inter- 
mediate body  a?,  such  that  if  A  strikes  it  with  a  given  velo- 
city a,  the  motion  communicated  through  x  to  G  may  be  a 
maximum." 

Because  Aa  is  the  momentum  of  A  before  impact,  it  re- 


110  EXAMPLES. 

suits,  from  the  laws  of  collision  of  elastic  bodies,  tliat 
— ^ is  tlie  velocity  of  x  after  impact ;  and  in  tlie  same 

way  X  communicates  the   velocity   ^-j --. ^r-    to  C, 

^  ^    {A+x)  {x  +  C)  ' 

which  must  be   a  maximum;   consequently,  its  reciprocal 

4Aaaj                   4Aa          V            x  J 
must  be  a  minimum,  which  clearly  requires  x  -f to  be  a 

minimum.      By  putting  the    differential   of   this  equal  to 

AC  .        .  

naught,  we  get  1  —  ~^~  =  0 ;  which  gives  x  =    V^AC,  the 

same  result  that  the  process  explained  on  page  101  will  give. 

Kemarks. — 1.  Putting  -j-  =  — r—  =  y  -r  —  m  =  the 

ratio  of  a?  to  A ;  we  readily  get  the  geometrical  progression 
A,  7/iA,  m^A^  771^ A,  &c. ;  which  may  be  supposed  to  result, 
as  in  the  question,  from  the  communication  of  motion  from 
A  through  mA  to  fnrA ;  and  from  mA  through  m^A  to  m^A ; 
and  so  on,  to  any  extent 

2.  It  is  also  clear,  that  a,  :; ,  -jz -r,  7- -,  &c., 

'  '  1  4-  m    (1  4-  ^0     (X  +  ^^^) 

are  the  velocities  of  the  successive  bodies,  which  are  also  in 
geometrical  progression  ;  and  that 

Aa.     zr-, Aa,        , )  Aa,     (- — —]  Aa,   &;c., 

'1  +  771  '        \l+77l/  '       \1  +  7/i/  '  ' 

also  in  geometrical  progression,  severally  express  the  mo- 
menta of  the  bodies.  Hence,  if  7?i  ==  1,  or  if  the  bodies 
equal  each  other,  they  will  have  equal  momenta ;  while,  if 
the  bodies  are  unequal,  their  momenta  will  increase  or  de- 
crease, accordingly  as  7?i  is  greater  or  less  than  unity. 


EXAMPLES.  Ill 

3.  As  a  result  of  these  principles,  it  has  been  proposed  to 
construct  speaking  and  hearing  trumpets  that  shall  be  more 
effective  than  those  previously  used. 

Thus,  let  the  trumpet  be  a  tube,  such  that  its  section  at 
right  angles  to  its  length  shall  always  be  a  circle,  and  that, 
when  the  distances  of  the  sections  from  the  place  of  the 
mouth  Or  ear  increase  in  arithmetical  progression,  the  radii 
of  the  sections  shall  increase  in  geometrical  progression ; 
then,  if  x  represents  the  distance  of  any  section  from  the 
place  of  the  mouth  or  ear  and  y  the  radius  of  the  section, 
we  may  express  the  connection  between  x  and  y  by  the 
equation  log  y  ==  a?,  or,  by  taking  a  for  the  base  of  the 
logarithms,  we  shall  have  y  =  a'',  the  well-known  equation 
of  the  logarithmic  curve;  whose  revolution  around  its  axis 
or  the  axis  of  a?,  which  is  an  asymptote  to  the  curve,  will 
generate  the  proper  figure  or  form  of  the  trumpet. 

4.  Now,  the  trumpets  being  filled  with  the  air,  which  is  a 
very  elastic  substance,  when  they  are  used  either  in  speaking 
or  hearing,  it  is  clear  that  the  sections  of  the  tubes,  regarded 
as  of  very  small  equal  thicknesses,  may  be  supposed  to  com- 
municate motion  successively  to  each  other,  like  the  elastic 
bodies  described  above ;  so  that  there  will  be  an  increase  of 
momentum  from  the  less  to  the  greater  sections  in  the  speak- 
ing trumpet,  and  a  decrease  of  momentum  in  proceeding 
from  the  greater  to  the  less  sections  in  the  hearing  trumpet. 

5.  It  is  easy  to  perceive  that  there  will  be  equal  momenta 
communicated  from  section  to  section,  through  a  prismatic, 
or  cylindric  column  of  air;  noticing,  that  the  sections  are 
always  supposed  to  be  perpendicular  to  the  lengths  of  the 
columns. 

18.  Supposing  the  ecliptic  to  be  a  circle,  it  is  required  to 
find  when  the  equation  of  time  is  a  maximum. 


112  EXAMPLEa 


Let  AB  and  AC  stand  for  tlie  ecliptic  and  equator, 
regarded  as  great  circles  of  the  celestial  sphere,  A  the  first 
point  of  Aries  and  S  the  place  of  the  sun  when  the  equation 
of  time  is  a  maximum ;  then,  the  arc  of  a  great  circle  of  the 
(celestial)  sphere  drawn  from  the  sun  perpendicular  to  the 
equator  meeting  it  in  S',  gives  AS,  AS',  and  SS'  for  the 
sun's  longitude,  right  ascension,  and  declination  (supposed 
north,  for  simplicity);  now,  since  time  is  reckoned  by  the 
sun's  motion  from  west  to  east  or  in  the  direction  of  the 
equator,  or,  which  comes  to  the  same,  because  S  and  S'  are 
on  the  same  celestial  meridian,  it  is  clear  that  the  time 
shown  by  the  sun  at  S  must  be  the  same  as  if  it  was  placed 
at  S' ;  whereas,  if  the  angle  A  =  0,  or  the  ecliptic  coincides 
with  the  equator,  the  point  S  will  be  reduced  to  the  equator 
so  that  AS  will  represent  the  sun's  right  ascension ;  and  of 
course  AS  —  AS',  when  reduced  to  time  at  the  rate  of  15° 
to  the  hour,  will  be  the  equation  of  time. 

From  spherical  trigonometry  we  have 

1  :  cos  A  : :  tan  AS  :  tan  AS'  =  tan  AS  x  cos  A ; 
consequently,  putting  tan  AS  =  x^  and 
c  =  cos  A  =  cos  23°  28'  very  nearly,  we  have  tan  AS'  =  ex. 
Because  when  AS  —  AS'  is  a  maximum, 

/  A  CI       A  o  /N       t3,n  AS  —  tan  AS'       x  —  ex      (1  —  c)  x 

tan  (AS  —  AS  )  =  - — —^ r-^  =  ^ —  \ '-^ 

^  ^      1+tanAStanAS       l  +  c-ar^       \ -^  Cdi? 

is  also  clearly  a  maximum,  it  is  clear,  since  c  is  invariable, 


EXAMPLES. 


113 


1  -\-  cx^       1 
tliat =  — h  ex  will  be  a  minimum ;   consequently 

OS  X 

(see  page    101),   we   shall  have   x  =  y-=  the  tangent 

of  the  sun's  longitude,  and  ex  =  the  tangent  of  the  sun's 
right  ascension  =  ^c,  when  the  equation  of  time  is  a  maxi- 
mum. Because  x  x  ex  =  y  -  x  \/e  =  1  =  V,  it  is  manifest 
that  we  must  have  AS  +  AS^  =  an  arc  of  90° ;  and  it  is 
clear  from  x  =  y-  ==  1.04416,  the  tan  of  46°  14',  that  the 

sun's  longitude  is  46°  14',  and  of  course  90°-46°  14'=::r43°  46' 
must  equal  his  right  ascension  when  the  equation  of  time  is 
a  maximum.  By  subtracting  AS'  from  AS  we  get  2°  28' 
for  the  maximum  value  of  the  equation  in  degrees,  &c., 
which  being  converted  into  time  at  the  rate  of  15°  to  the 
hour,  and  15'  to  the  minute,  &c.,  gives  9  minutes  and  52 
seconds,  for  the  maximum  value  of  the  equation  of  time. 

19.  Supposing  the  frustum  of  a  right  cone  whose  base  and 
altitude  are  given,  to  move  forward  in  a  resisting  medium  in 
the  direction  of  its  length,  having  its  lesser  end  foremost ;  to 
find  the  diameter  of  the  lesser  end,  when  the  resistance  of 
the  medium  is  a  minimum. 


114  EXAMPLES. 

Let  BCFG  represent  the  frustum,  BO  =  a  and  G  =  a*  D 
the  radii  of  the  greater  and  lesser  ends,  and  OD  =  A  the 
height  The  frustum,  moving  in  the  direction  OD,  it  is 
plain  that  the  resistance  of  the  medium  will  act  in  a  com- 
trary  direction,  so  that  the  line  7n,p  parallel  to  OD  may  stand 
for  the  resistance  of  a  particle  of  the  medium,  which  by 
drawing  mq  perpendicular  to  the  side  of  the  frustum,  may 
be  resolved  into  the  forces  mq  and  qp,  of  which  the  force  //?// 
is  alone  to  be  retained,  since  qp  acting  in  the  direction  of  the 
slant  side  of  the  frustum,  can  not  sensibly  affect  its  motion ; 
and,  in  like  manner,  by  drawing  qn  at  right  angles  to  7np, 
we  may  separate  the  force  mq  into  the  two  forces  mn  and 
nq,  of  which  mn  is  alone  to  be  retained,  since  nq  is  evi- 
dently destroyed  by  an  equal  and  opposite  force. 

Hence,  if  pm  is  represented  by  unity  or  (1),  it  is  plain 
that  7/i«,  the  only  effective  part  of  7n.p,  must  be  represented 
by  the  square  of  the  sine  of  mpq,  the  angle  of  incidence  of 
the  resisting  particle,  with  the  side  of  the  frustum. 

FL  being  parallel  to  mp,  we  may  clearly  take  the  angle 

CFL,  whose  sine  equals  CL  -^  OF  =     „.,, — ; r,^ ,  for 

^  i/lh^  -{-{a  —  a.')'} 

(a  _  xf 
the  angle  of  incidence  ;  consequently,  -j^ — .  ■         .,     equals 

ft  -J-  \(i  —  3?y 

the  resistance  of  each  particle  that  strikes  the  slant  surface  of 
the  frustum,  while  unity  equals  the  resistance  of  each  parti- 
cle against  the  smaller  end  of  the  frustum. 

If  sir  represents  the  number  of  particles  that  strike  the 
lesser  end  of  the  frustum,  it  is  plain  that  a-  —  ar*  will  repre- 
sent the  number  of  particles  that  strike  the  curve  surface  of 
the  frustum. 

Hence,  since  the  resistance  of  each  particle  against  the 
lesser  end  of  the  frustum  is  perpendicular  to  it  and  repre- 


EXAMPLES.  115 

sented  hj  unity,  it  is  plain  that  x-  may  be  taken  for  the  re- 
sistance   against    the    lesser    end    of    the    frustum,   while 

,   ^ — rr    X  (a^  —  x^)  represents  the  resistance  apjainst 

/r  -\-  {a  —  xf        ^  ^      ^  ^ 

the  curve  surface  of  the  frustum ;  consequently 
^2        {a  —  xf  (a^—  x^)  _   AV  +  a}  {a—xj- 

2         (a?-  — a^)A2 


'    ia-xf  +  h'' 

=  the  resistance  to  the  whole  surface  of  the   frustum  =  a 

minimum. 

x^  —  a- 
It  is  hence  clear  that  -, ri; tt,  must  be  a  minimum, 

(a  —  x)'  +  A- 

,  (a  -  xf  -{-  h'  - 

or  its  reciprocal  ^ — -^—^ — ^ —  must  be  a  maximum. 
x"  —  a 

By  putting  the  differential  of  this  equal  to  naught,  we 
readily  get  the  equation  ^^ — — --  =  — ,  whose  solution  gives 


2a'  +  h'  T  A  ^4:a'  +  /r 
2a  . 


.     ,  .  ,                       2a'  +  /i'-hV4.a'  +h' 
of  which,  X  = 


2a 

is  clearly  the  only  root  that  is  applicable  to  the  question, 
since  the  other  root  will  be  greater  than  the  radius  of  the 
base  of  the  frustum.     Erom 

{a  -  xf         h'        ^         (a  -  xf        X       CU       DF 

=  —  we  have  ^ — j-^-^  =  -  or  :i=,^,,  =  7-^7^-  ; 

X  a  li"  a       FL-       CO  ' 

which,  supposing  the  cone  completed,  as  in  the  figure,  gives 

1^    CO'-        SD 

H   o7jo   =  ^7^  or  its  equivalent  CO^  1=  SO  x  SD. 

Wk       Hence,  bisect  OD  in  Q  and  join  QO,  and  set  QO  from  Q 

I 


116  EXAMPLES. 

to  S'  on  QD  produced ;  then,  from  the  right-angled  triangle 
CQO  we  have 

CO''  =  CQ^  -  QO^  =  QS'2  -  QD' 

=  (QS'  +  QD)  (AS'  -  QD)  =  OS'  x  S'D, 

as  it  ought  to  do ;  consequently,  the  trapezoid  CODF'  revolv- 
ing about  OD  as  an  axis,  will  clearly  generate  the  frustum  of 
minimum  resistanca 

Kemarks. — 1.  It  will  be  perceived  that  the  frustum  BCFG, 
has  been  taken  at  hazard,  and  thence  from  the  reasoning  the 
equation  CO^  =  SO  x  SD  found,  which  has  enabled  us  to 
find  the  true  frustum,  as  above. 

2.  We  have  taken  the  example  from  the  scholium  to  Prop. 
34,  Sec.  7,  Vol.  2,  of  Newton's  "Principia;"  and  it  is  easy  to 
perceive  that  the  preceding  construction  is  the  same  as  that 
of  Newton. 

20.  To  find  the  position  of  Yenus  when  brightest,  sup- 
posing the  orbit  of  the  earth  and  that  of  Yenus  to  be  circles 
in  the  same  plane,  having  the  sun  in  their  common  center. 

Let  S,  E,  Y,  denoting  the  centers  of  the  sun,  earth,  and 
Yenus,  be  connected  by  right  lines  forming  a  triangle ;  repre- 
senting the  sides  SE,  SY,  and  EY,  by  a,  ^,  and  a?,  and  using 
the  circle  abed  to  represent  a  section  of  Yenus  by  the  trian- 
gle SYE  ;  then,  ac  and  hd  being  diameters  of  Yenus  perpen- 
dicular to  its  distances  SY  and  EY  from  the  sun  and  earth, 
it  is  manifest  that  ah  may  be  taken  to  represent  the  breadth 
of  the  illuminated  part  of  Yenus,  which  by  drawing  az  per- 
pendicular to  hd  gives  hz  for  the  versed  sine  of  the  angle 
oYh^  when  the  radius  of  Yenus  is  represented  by  unity, 
which  may  clearly  be  taken  to  vary  as  the  part  of  Yenus 
that  reflects  light  to  the  earth  at  E ;  consequently,  from  the 


EXAMPLES. 


117 


principles  of  optics    -pttti  will  express  the  quantity  of  light 
Hi  V 

reflected  by  Venus  to  the  earth. 


From  the  triangle  SEV,  we  have 
SE2=SY^+Ey^-2SYxEVcos  SYE=:5^+ar'+25^  cosaVJ 

since  the  sum  of  SVE  and  oNh  is  clearly  equal  to  two  right 
angles. 

Eepresenting  cos  aVJ  by  cos  0,  since  SE^  =  a-^  we  readily 

(j^  _  })^  _  x^ 

get  from  the  preceding  equation  cos  0  = ^7 ,  which 

1  —  cos  0      Qy-^xf  —  G^ 
gives  —7^-  = -iw-  = 

a  maximum,  since  1— cos  </>  represents  the  versed  sine  of  the 
angle  aNh.     Since 

ij}  +  xf  —  a^_{b  +  x  +  a) .  (5  +  x  —  a)  _ 


2bj 


llx 


118  EXAMPLES. 

a  maximum,  by  taking  the  hyperbolic  logarithm  of  this,  we 
must  have 

log  {b  -\-  x  +  a)  +  \og(h  +  x  —  a)  —  S  log  x  —  log  2h  = 
a  maximum ;  consequently,  putting  the  differential  of  this 
equal  to  naught,  we  shall  have  (see  p.  54) 

h -^  X  +  a  b  +  x^a  x  ' 
or  its  equivalent  —  Sh^  —  4:bx  —  a^  -{- Sa^  =  0 ;  consequently 
solving  this  quadratic,  we  have  x=  —2b  -\-  Vb'  +  Sa^  and 
x=:  —2b—  Vb^  -f  3a-,  which,  giving  x  negative,  must  be  re- 
jected, and  of  course  we  shall  have  x=  —2b  ■{-  Vb'  +  Sa\ 
By  representing  a  and  b  by  their  proportional  distances 
1  and  0.72333  nearly,  we  get,  from  tlie  preceding  equation, 
X  =  0.4304:6  for  the  distance  of  Venus  from  the  earth. 
Hence,  we  easily  get  SEY  =  39°  44'  for  the  elongation  of 
Yenus  seen  from  the  earth,  and  ESY  =  22°  21',  the  elonga- 
tion of  Yenus  seen  from  the  sun,  which  being  less  than 
43°  40',  Yenus's  greatest  elongation,  shows  that  she  is  bright- 
est between  her  greatest  elongation  and  her  inferior  conjunc- 
tion, being  nearly  half  way  between  the  inferior  conjunction 
and  greatest  elongation. 

Because  the  preceding  reasoning  does  not  give  the  posi- 
tions of  Yenus  when  she  reflects  the  minimum  light,  we 
shall  determine  these  positions  after  the  following  method. 

Thus,  from  p.  95,  we  have 

¥{x-  h)  =  F^  -  '^^  h  +  fP  (F.x)  ^  -,  &c., 

and    F(.  +  /0  =  F.4--A_JA+_A_^__+,&e.; 
where  it  will  be  noticed,  that  we  have  shown  Yx  can  not  be  a 


EXAMPLES.  119 

maximnm  or  minimum,  unless  it  is  determined  on  the  sup- 
position that  the  term  —^ — -  h  is  made  to  disappear  from  the 

equations.  Now  it  is  easy  to  perceive,  that  we  have  thus  far 
made  the  term  disappear  from  the  equations  by  assuming 
d  (F,r) 


dx 


0 ;  we  now  observe  that  we  may,  when  necessary, 


make  the  term  disappear  from  the  equations  by  putting  A  =  0, 

•        ^     c^(F.r;),  ..     ^.         .      d{Fx). 

or,  since  lor  -~ — -n^  we  may  evidently  w^rite  — ^ — -  dx,  we 
ctx  CCX 

may  assume  dx  =  0;  which,  in  this  question,  clearly  indi- 
cates the  inferior  and  superior  conjunctions  of  the  planet ; 
since  x  is  a  minimum  and  maximum  at  those  points. 

Kemabks.— 1.  By  putting  h  =0.3871,  we  find  x  =1.00058, 
and  thence  get  SEV  =  22°  19'  for  the  elongation  of  Mercury 
when  brightest.  Also,  the  angle  ESY  =  78°  56',  while  it  is 
only  67°  13 '.5  at  the  time  of  the  planet's  greatest  elonga- 
tion ;  consequently  Mercury  is  brightest  between  its  greatest 
elongation  and  the  superior  conjunction. 

2.  Because  the  motion  of  Venus  about  the  sun  relatively 
to  that  of  the  earth  is  about  37' ;  by  dividing  22°  21'=1341' 
by  37',  we  get  36  days  for  the  time  when  Venus  is  brightest 
before  and  after  her  inferior  conjunction. 

3.  If  we  apply  the  formula  x  =  —  2h  -{-  Vh'-i-  3a^,  to  find 
the  j)Osition  of  a  superior  planet  when  brightest,  it  w^ill  be 
found  to  be  impossible ;  for,  since  Scr  will  be  less  than  Sir, 
it  follows  that  VP  +  3a^  will  be  less  than  2h,  and,  of  course, 
X  =  —  2h  +  Vh^  +  3(x^  will  be  negative,  which  is  impos- 
sible, since  x,  the  distance  of  the  planet  from  the  earth,  is 

always  positive.     Hence,  it  is  manifest  that  — ^—  dx  =  0, 

CiX 

when  applied  to  the  superior  planets,  can  only  be  satisfied 


120 


EXAMPLES. 


by  putting  dx  =  0;  which  clearly  indicates  that  they  reflect 
the  most  light  in  their  oppositions,  and  the  least  in  their 
conjunctions  with  the  sun. 

21.  From  the  extremity  of  the  minor  axis  of  an  ellipse 
to  draw  the  maximum  line  to  the  opposite  part  of  its  pe- 
rimeter. 


Let  AEB  be  an  ellipse,  having  AB  and  DE  for  its  major 
and  minor  axes  ;  and  let  EF  be  the  maximum  line  required, 
having  EG  and  FG  for  the  rectangular  co-ordinates  of  its 
extremity  F.     Then,  a  and  h  representing  the  major  and 

2 

minor  axes,  we  have  ¥  :a^  : :  hx  —  ay^ :  y^  =-j^  (hx—x%  from 
a  well-known  property  of  the  curve.  Hence,  adding  sc^  to 
y',  we  have  E'F'^EG' +  ¥0^  =  0^ -i- ^  =  0^ +-^,{hx-x^); 

consequently,  gince  EF^  is  a  maximum,  ar^  +  -to  {^^^  ~  ^) 
must  be  a  maximum ;  and  putting  its  differential  equal  to 

naught,    we    have     2a?  -j-  -y^  (J  —  2x)  =  0,    which    gives 
1     a'h 


^  =  9  "2 — Jfi  1     which    is    clearly   a   maximum,   since  the 

differential  of  2aj  +  j^  {b  —  2x)  is   (2 v^  1  dx,   which  is 

negative,  when  dx  is  positive,  as  it  ought  to  be. 


EXAMPLES.  121 

Eemarks. — 1.  It  is  evident  from  the  nature  of  the 
ellipse,  that  the  relation  of  its  axes  must  be  such  that  x 
shall  not  be  greater  than  the  minor  axis  J,  in  order  that  the" 
preceding  value  of  x  may  be  applicable  to  it ;  since  EG  must 
clearly  not  be  gi-eater  than  ED  —  K 

Hence,  to  find  the  greatest  value  that  h  can  have  when  the 

preceding  solution  is  possible,  we  put  h  ior  x  m  x  =  ^  -^ — p, 

and  thence  get  2¥  =z  a^  or  h  =^—^ ;  consequently,  when 

1)  has  this  or  a  less  value,  the  maximum  line  wiU  be  found 

from  x=z^  —^ 75. 

Z  a^  —  0- 

The  question  and  the  substance  of  what  has  been  said 
are  substantially  the  same  as  given  by  T.  Simpson,  at  pages 
35  and  36  of  his  "  Fluxions,"  for  the  purpose  of  showing 
whether  the  solution  found  in  any  case  falls  within  the 
limits  required  by  the  nature  of  the  question, 

a- 
2.  Eesuming,  EF^  =  x^  -\-~  (px  —  x%  and  putting  its  dif- 
ferential equal  to  naught,  we  have  l2x  +'jt{^  —  2x)\  dx=0; 

which,  by  putting  dx  =  0,  clearly  gives  the  minor  axis  ED 
for  the  maximum  line  when   the   minor  axis   is   not  less 

than  — -  .  When  the  minor  axis  is  not  greater  than  — ,  by 
putting  the  factor  2x  +  jj{h  —  2x)  of  the  preceding  differen- 
tial equal  to  naught,  we  get  x=  -  — r^,  to  determine  the 

maximum  value  of  the  line  to  be  drawn ;  and  by  putting 
dx  =  0,  we  get  the  minor  axis  ED  for  the  minimum  value, 
as  is  manifest  from  the  consideration  that  when  h  is  less 

6 


122  EXAMPLEa 

than  ^  tliere  will  be  two  maxima  values,  represented  by  EF 

and  EF',  and  of  course  ED  must  be  a  minimum. 

It  is  lience  clear  that  the  (common)  rule  for  finding  the 
maxima  and  minima  of  a  function  of  a  single  variable,  given 
at  p.  96,  is  not  always  sufficiently  general 

22.  Given         x  -\-  y  -\-  s  =  a    and    x^ifz^ 
or  m  log  X  +  n  log  y  -r  p  log  {a  —  x  —  y) 

a  maximum,  to  find  a?,  y,  and  z. 

It  is  manifest  that  (since  x  and  y  are  independent  varia- 
bles) we  may  put  the  differential  of  the  preceding  equation, 
with  reference  to  y,  equal  to  nothing,  and  thence  get  y  in 
terms  of  a? ;  by  which  means  we  shall  reduce  the  question  to 
that  of  making  a  function  of  a  single  variable  equal  to 
naught 

Thus,  we  shall  have 

2^__^^^=:0     or    ^ ^— =0, 

y         a  — x—y  y       a— x—y 

which  gives  y  =  — ;  consequently,  putting  this  value 

for  y  in  771  log  x  -^  n  log  y  +p  log  {a  —  x  —  y\  it  becomes 
mlogx  i-n  log (a-x)  +  log  ?i" 
—  (p  +  n)  log  {n  -hp)  4-i?  log  (a  —  a?)  +  log^^, 
Vv'hich  must  be  a  maximum ;  consequently,  putting  its  dif- 
ferential equal  to  naught,  we  have 

\x  a  —  x  I 

whose  diilerential  gives 

\x^         {a  —  xyj 


EXAMPLES.  123 

From  the  first  of  these  equations  we  get 

am 

and  the  second  equation  shows  x"^y'^z^  to  be  a  maximum,  as 
required. 

Eemaeks. — 1.  This  example  has  been  before  solved  at 
page  100,  and  it  is  easy  to  perceive  that  we  obtain  the  same 
results  as  there  found,  by  substituting  the  value  of  x  in  that 
of  ?/,  and  then  substituting  the  values  of  x  and  y  for  them, 
in  2  =  a  —  x  —y. 

2.  We  have  here  given  the  preceding  solution  of  it,  for 
the  purpose  of  showing  the  facility  with  which  a  function  of 
any  number  of  independent  variables  may  be  made  a  maxi- 
mum or  minimum,  by  reducing  it  to  the  maximum  or  mini- 
mum of  a  function  of  a  single  variable ;  since  it  is  easy  to 
perceive  that  we  may  proceed  in  like  manner,  whatever  may 
be  the  number  of  independent  variables. 

To  make  what  is  here  said  more  clear,  we  will  apply  the 
procTess  to  the  following  example  from  page  33  of  Simpson's 
"Fluxions." 

23.  To  find  such  values  of  a,',  y,  and  s,  as  shall  make 
ij/  —  x^)  {x"2  —  B^)  (xy  —  y'^)  a  maximum. 

From  making  y  alone  variable,  we  have  xdy  —  2ydy  =  0  ; 

o 

X  X" 

which  gives  ?/  =  ^  ,  and  thence  xy  —y'^  ^'-r  . 

In  like  manner,  making  z  alone  variable  in  the  proposed 

X 

equation,  we  have  x'dz  —  Sx^dz  —  0  ;  which  gives  z  —  —^^ 

4/0 

and  thence  x^z  ~  z^  =  - — -■ .     By  substituting  the  preceding 
values  in  the  proposed  expression,  it  becomes 


124  EXAMPLES. 

(¥  —  ar^)x  TT-^   X  7-  =  maximum ; 

consequently,  we  must  make  &V  —  a?^  a  maximum. 

Hence  we  "have  6h^x*dx—8ju^dx=0,  and  {20h^x^—oQxyix'; 

tlie  first  of  these  gives  ^=  ^  |/5,  which  put  for  x  in  the  sec- 
ond expression  makes  it  negative,  and  of  course  shows  the 
proposed  expression  to  be  a  maximum  as  required. 


SECTION  y. 


TANGENTS     AND     SUBTANGENTS,     NORMALS     AND     SUBNOR- 
MALS,  ETC. 


M  0    N 


(1.)  Suppose  AM  =  x  and  MS  =  y  to  stand  for  the 
abscissa  and  ordinate  of  the  point  S  of  the  plane  curve  ASY, 
having  A  for  their  origin  ;  then,  conceiving  the  curve  to  be 
described  by  the  extremity  S  of  the  ordinate,  while  the  other 
extremity  M  moves  uniformly  from  the  origin  A  of  the  co- 
ordinates along  the  line  of  the  abscissae  toward  P,  so  as  to 
keep  the  ordinate  constantly  parallel  to  itself  during  the 
motion,  we  may  clearly  suppose  the  ordinate  to  increase  or 
decrease  in  such  a  way  as  to  describe  the  curve. 

(2.)  If  now  we  suppose  the  right  line  TS^  to  pass  through 
the  extremity  S  of  the  ordinate  of  the  curve  AS  V  in  any  one 
of  its  positions,  in  such  a  way  that  the  first  differential  coeffi- 
cient of  the  equation  of  the  curve  equals  the  first  differential 
coefficient  of  the  right  line ;  then,  the  right  line  is  said  to 
touch  the  curve^  or  the  line  and  curve  are  said  to  touch  each 
other ^  and  the  point  S  is  called  their  points  of  contact. 

Thus,  if  y  =  0  {x)  and  y  =  Ax  +  B  represent  the   equa- 


126  DRAWING  TANGENTS,    ETC. 

tions  of  tlie  curve  and  right  line,  bj  taking  their  differential 

coefficients,  we  shall  have  -4-  =  — >—  and  -~  =  A:   conse- 

dx         ax  ax 

d<b  (x\ 
qnently,  according  to  the  definition,  we  shall  have  A  =  — ,~ ; 

which,  when  x  is  known,  will  give  the  value  of  A,  and 
thence  the  tangent  can  easily  be  drawn. 

Since  A  =  -^  is  derived  from  the  equation  of  the  curve, 
ax 

if  X  and  y  represent  the  co-ordinates  of  the  point  of  contact 

of  any  tangent  with  the  curve,  and  X  and  Y  the  co-ordinates 

of  any  other  point  of  the  tangent,  it  is  manifest  that 

Y-,=  |(X-.) 

may  be  taken  as  the  general  equation  of  the  tangent. 
Putting  Y  =  0,  in  this  equation,  we  easily  get 

-f^  =  aj-X  =  MT, 
ay 

dx 

dij 
called  the  suhtangent^  which  is  known,  since  y  and  -—-  are 

ax 

known  from  the  equ.ation  of  the  curve ;  noticing,  since  x  and 
y  are  supposed  to  be  taken  in  the  direction  of  the  positive 
co-ordinates,  that  X  =  —  AT  taken  in  the  direction  of  x 
negative,  must  be  subtracted  from  x  =  AM  taken  in  the  di- 
rection of  X  positive,  which  gives  a?  —  X  =  AM  -f  AT  =  MT, 
as  above. 

It  may  be  proper  to  illustrate  what  is  here  done,  in  another 
way :  thus,  let  the  ordinate  of  the  right  line  EG  be  drawn 
very  near  SM,  SQ  be  drawn  parallel  to  MO  or  the  axis  of  x ; 
then,  representing  MO  or  SQ  by  dx,  it  is  clear  that  QR  will 
represent  dy  both  in  the  right  line  and  curve ;  and  that  the 


TANGENTS,    ETC.  127 

triangles  SRG  and  STM,  being  equiangalar,  give  (from  well- 
known  principles  of  geometry)  the  proportion 

KQ  :  SQ::SM  :  MT, 

or  its  equivalent  -j-  —  SM  ~  MT ;  which  gives  ~  —  TM,  or 

dx 

-~-  =  MT,  which  agree  with  the  expression  for  the  subtan- 
d?/ 

gent  before  found.  Where,  it  may  be  added,  that  if  the  co- 
ordinates are  at  right  angles  to  each  other, 

^  =  tan  ang  ESQ  =  tan  ang  STM 

to  radius  unity ;  consequently,  we  find  the  subtangent  hy 
dividing  the  ordinate  by  tJie  tangent  of  the  angle  of  the 
inclination  of  the  tangent  to  the  line  of  the  ahscissce,  when 
the  axes  of  the  proposed  curve  are  rectangular,  or,  which 
comes  to  the  same,  we  multiply  the  ordinates  hy  the  tangent 
of  the  angle  it  makes  with  the  tangent^  for  the  suhtangent. 

(3.)  If  SIST  is  drawn  through  the  point  of  contact  S,  per- 
pendicular to  the  tangent,  meeting  the  axis  of  x  in  N",  it  will 
be  what  is  called  the  normal ;  particularly  when  AP  is  the 
axis  of  the  curve,  or  cuts  the  tangent  at  A  perpendicularly, 
and  the  ordinates  of  the  curve  are  perpendicular  to  the  axis  ; 
also,  MN  is  called  the  subnormal. 

Since  the  triangles  SRQ  and  STM  are  equiangular,  it  is 
manifest  that  whatever  may  be  the  angle  AMS,  provided  it 
is  known,  we  can  always  find  all  the  parts  of  the  triangle 
STM  from  the  equation  of  the  curve   and  knowing  the 

ordinate  SM,  since  SM  —  y  gives  TM  =  y^-j^=  the  sub- 

tangent.     Hence,  we  easily  get  SN,  the  normal,  from  the 


128 

triangle  STN,   and  TN  from  the  same    triangle ;    conse- 
quently, TN  —  TM  =  MN  =  tlie  subnormal  is  found. 

We  shall,  in  what  is  to  follow  (according  to  custom),  sup- 
pose AP  to  be  the  axis  of  the  curve,  whose  ordinate  y  cuts 
it  perpendicularly ;  then,  the  triangle  SRQ  will  evidently  be 
similar  to  the  triangle  SMN,  and  will  give 

SQ  :  EQ  ::  SM  :  MN,  or  ^0?  :  ^y  ::  7/  :  MN  =  ^ 

and  thence  the  normal 

is  found:  noticing,  that  in  much  the  same  way  from  the 
triangle  STM,  we  get  the  tangent  ST  =  y  y    1  +  \  \-j-)  [• 

There  is  another  way  of  finding  the  subnormal  from  the 
equation  of  the  normal,  which  it  may  not  be  improper  to 
notice  in  this  placa 

Thus,  assuming  Y—  y  ==  A  (X—  a?)  to  represent  the  equa- 
tion of  the  normal,  we  shall  have  A  equal  to  the  tangent  of 
the  ang.  SNP  to  radius  unity;  observing,  that  the  angles 
which  right  lines  make  with  the  axis  of  a?,  are  supposed  to 
be  included  between  them  and  x  positive,  estimated  (accord- 
ing to  usage)  from  right  to  left. 

Because  the  angle  SNP,  from  well-known  principles  of 
geometry,  equals  the  sum  of  the  inward  and  opposite  angles, 
S  =  90°  and  T  of  the  triangle  NST,  we  shall  have 

+       ^^ix^-D         .       /oAo   ,   mx  sin  (90°  +  T)  cos  T 

tail  S^P  =  tan  (90°  +  T)    =   —7^3 ^^   =    -. — p^ 

^  ^         cos  (90°  +  T)  —  sm  T 

=  —  I — F?i  = 1-  = 5- ;  see  pp.  62  and  63. 

tanT  d7/  dy  ^^ 

dx 
Hence,  from  the  substitution  of  the  value   of  A  in  the 


TANGENTS,   WITH  AN  ILLUSTEATION.  129 

equation  of  tlie  normal,  it  will  become 

dx 
whicli  is  the  well-known  equation  of  tlie  normal. 

If  in  this  we  put  Y  =  0,  we  shall  have  —y=  —  --  (X.—x) ; 

or,  since  X  —  a?  =  A^  —  AM  ==  MN,  the  subnormal  =  ~^, 

dx 

the  same  as  before  found. 

(4.)  Kesuming  the  equations  that  have  been  found,  when 
the  ordinates  are  perpendicular  to  the  line  of  the  abscissae, 

we  shall  have  Y  ~  y  —  y-  {X  —  x) (1) 

for  the  equation  of  the  tangent  that  passes  through  the  point 
(x,  y)  of  any  plane  curve ;  and 

Y-,=  -l(X-.)  =  -|(X-.) (2) 

dx 
is  the  equation  of  the  normal  to  any  plane  curve,  at  the 
point  (-T,  y) :  noticing,  that  these  formulas  clearly  show  that 
to  find  the  tangent  or  normal  at  any  point  {x,  y)  of  any  plane 

curve,  it  will  be  necessary  to  find  y-  from  the  equation  of 

the  curve,  and  to  put  its  value  in  the  preceding  equations. 

Thus,  to  find  the  tangent  and  normal  to  the  circle  whose 
equation  is  y^  ^  x-  ^=.  y*^,  r  being  the  radius. 

By  taking  the  differential,  we  have  ^ydy  +  2xdx  =  0 ; 

which  gives  -^  =  —  - ;  and  substitutinsr  this  value  of  —- 
^         dx  y'  ^  dx 

X 

in  (1)  and  (2),  we  have  Y—y= (X  —  a?),  or,  by  a  simple 

6* 


130      TANGENTS  TO  THE  ELLIPSE,  HYPERBOLA,   ETC. 

reduction,  Yy  +  Xa?  =  y-  +  x-  =  r- ;  and  Y  —  y=^-Qs.  —  x\ 

or  Yx  .=  yX,  which  shows  that  the  normal  passes  tli rough 
the  center  of  the  circle  at  right  angles  to  the  tangent. 

Taking  a^y"^  +  h'jr  =  aVj^^  the  equation  of  the  ellipse, 
and  proceeding  as  before,  we  get  a^Yy  +  lrX.x  =  a-Z>-,  and 
a-Xy  —  ¥Yx  =.  {a^  —  Ir)  y.»,  for  the  equations  of  the  tangent 
and  normal,  which  are  well-known  forms. 

Supposing  a  and  h  to  be  the  half  major  and  minor  axes 
of  the  ellipse,  bj  putting  Y  and  X  successively  equal  to 
naught  in  the  preceding  equations,  we  have 

X  =  —  and  Y  = — ,     and    X  — r —  x.     Y  = -nr  y ; 

which  are  well-known  forms  for  drawing  tangents  and  nor- 
mals to  the  ellipse. 

Kemarks. — 1.  Because  ay  —  h^x^  =  —  a/U^^  the  equation 
of  the  hyperbola,  is  deduced  from  that  of  the  ellipse  by 
changing  the  signs  of  its  terms  that  involve  h- ;  it  is  clear 
that,  by  cha,nging  the  signs  of  the  terms  that  involve  Ir  in 

the  above  results,  they  will  give    X  =  —  and  Y  =  , 

X  y 

and   X  = 2 —  a?,     Y  =  — ry —  y,  for  the  corresponding 

(X  0 

quantities  in  the  hyperbola, 

2.  From  X  =  —  and  Y  = ,  by  supposing  x  and  y  to 

X  y 

be  infinitely  great,  it  is  clear  that  X  and  Y  will  become  un- 

limitedly  small,  or  that  tangents  to  the  hyperbola  at  points 

infinitely  remote  from  the  center  will  pass  through  it  very 

nearly,  and  have  ay  —  J V  =  0,  or  its  equivalents  y  =  -  x 

Ob 

and  y  = a?,  the  equations  of  right  lines  passing  through 


TANGENTS  WITH   HYPERBOLIC  ASYMPTOTES. 


131 


tlie  center  of  tlie  hyperbola,  for  their  limits ;  in  sncli  a  sense, 
that  they  will  ultimately  differ  insensibly  from  coinciding 
with  these  equations:  noticing,  that  these  limits  are  called 
the  asymptotes  of  the  hyperbola ;  and  from  <rif-  —  lr£-  =  0 
and  (ry-  —  V^Ji-  =  —  (i-l)^^  it  is  also  clear  that  the  asymptotes 
may  be  regarded  as  exterior  limits  of  the  hyperbola. 

3.  It  is  easy  to  perceive  that  the  preceding  conclusions  with 
reference  to  tangents  and  asymptotes  are  independent  of  the 
angle  formed  by  the  axes  of  co-ordinates ;  provided  a  and 
h  represent  a  pair  of  semiconjugate  diameters  having  the 
same  directions  as  the  axes  of  co-ordinates. 

(5.)  We  will  now  show  how  to  draw  tangents  and  normals 
to  plane  curves,  referred  to  polar  co-ordinates. 


Thus,  let  T'S  ==  z'  be  the  radius  vector  drawn  from  the 
pole  T  to  any  point  S  of  the  curve,  ArP  the  angular  axis 
making  the  angle  P/'S  =  w  with  the  radius  vector ;  then 
through  the  pole  draw  Nz-Q  at  right  angles  to  the  radius 
vector,  meeting  the  tangent  to  the  curve  at  S  in  Q  and  the 
normal  SN  to  the  curve  at  the  same  point  in  N ;  then  we 
shall  take  rQ  and  rl^  for  th:)  sub  tangent  and  subnormal  of 
the  tangent  SQ  and  normal  SN  to  the  curve  at  S.  Drawing 
SM  =  y  perpendicular  to  the  angular  axis  AP,  we  may 
evidently  take  /'M  =  x  and  SM  =  y  for  the  rectangular  co- 
ordinates of  the  point  S  of  the  proposed  curve,  having  r 


132  TANGENTS  AND  POLAR   CO-ORDINATES,    ETC. 

for  their  origin;  and  from  the  right-angled  triangle  ?SM, 

we  get,  from  the   well-known  principles   of  trigonometry, 

T  sin  CO  =  y  and  r  cos  w  =  a?,  which  give  7*^  =  ar^  -f  if. 

Because  the  angle  SrP  =  w  is  the  exterior  angle  of  the 

triangle  ST/*,  we  shall  have  the  angle  S  =  w  —  T,  which  gives 

X      Q        i.      /  rn\       tan  0)  —  tan  T  .         ^  y 

tan  S  =  tan  (w  —  T)  =  - — — j^  ,  or  since  tan  w  =  ^ 

^  ^       1+tanwtanT  x 

and  tan  T  =  -TT^  (page  127),  we  have 

\  X         dxl        \         xdxl        xdx  +  ydy 


_"  y 

r^  sin^  0)  c?  cot  w 

rdo) 

~\d^~ 

rdr            ~ 

~~dF' 

Hence,  since  tan  S 

= -r-  and  cot  S 

dr 

\  =  —  tan  rSN,  we 

get  from  the  triangles  ?'SQ  and  rSN,  by  1 

brigonometry, 

rQ  =  Sr 

X  tan  S    or    rQ  = 

r^d(D 
dr  ' 

and      T'N  =  ~  Sr  x 

cot  S  =  —  7*  -= r 

dw           dr 
dr "      di^' 

for  the  expressions  for  the  subtangent  and  subnormal,  as  re- 
quired. 

It  may  be  added,  that  if  we  regard  x  and  y  as  functions 

of  6),  we  may  for  dx  and  dy  in  tan  S  =  —^ ^,  evidently 

write  -^  and  ~- ;    consequently,    from    x  =:  r  cos   w   and 
di>i  ctw 

y  =  r  m)  w,  we  readily  get 

dx      dr  .  ,    dy      dr   . 

-^■=—-  cos  G)—r  sm  w    and    ^  =  -7-  sm  w  +  r  cos  w. 

aw      aw  aw      aw 

Hence,  from  the  substitution  of  these  values  of 


TAKGENTS  AND   POLAK   CO-ORDINATES,    ETC. 


133 


dx       ,  (ill .    ^      ^ 
-7-  and  —  m  tan  b  -- 

we  get,  after  obvious  reductions, 


ydx  —  xdy 
xdx  -\-  ydif 


r  tan  S  = 


dr 
dco 


r-dd) 


= r: — ,  and  thence  rN 

dr 


-7-,  the  same 


as  before;  see  p.  117  of  Young's   ''Differential  Calculus." 

Since  the  angles  SrP  and  SrT  —  180°,  if  we  represent 
Sz-T  bj  e,  we  shall  get  w  =  180°— 0,  which  gives  do)  =  —  dd. 
Hence,  the  preceding  equations  become 

'''        (3) 


r  tan  S  =  /"  X 


dr 


and 


r  cot  S  —  7*  tan  rSN"  = 


dr 
dd' 


(4): 


noticing,  that  0,  the  angle  T/'S,  is  the  length  of  an  arc  of  a 
circle  whose  radius  is  unity  and  center  r,  which  is  measured 
from  tT  toward  7'S,  and  may  contain  one  or  more  circum- 
ferences, or  any  part  of  the  circumference,  according  to  the 
nature  of  the  case. 

Remark. — The  substance  of  what  has  been  done  may  be 
expressed  in  the  following  simple  manner. 


Thus,  from  r  draw  the  right  line  rh  to  make  the  small 
angle  dd  with  rS,  and  from  /■  as  a  center,  with  radii  rm  =  1 
and  rS  =  r,  describe  the  small  arcs  mn  and  Sa,  cutting  rS 


134  TANGENTS  AND   POLAR  CO-ORDINATES,   ETC. 

and  rh  in  tlie  points  m.n  and  Sa ;  then,  mn  =  dd,  and  from 
the  similarity  of  the  circular  sectors,  we  have 
1  :  dd::r:  Sa  =  rdd. 

Then,  through  S  draw  So  perpendicular  to  rS  and  equal  to 

rdO^  and  ch'  perpendicular  to  Sc,  meeting  the  tangent  T^  in 

h' ;  then,  the  triangle  Scb'  is  evidently  equiangular  to  the 

triangles  rSQ  and  rSN.     From  the  equiangular  triangles 

Sob'  and  SrQ,  we  have  the  proportion  h'c  :  So  : :  Sr  :  rQ,  or 

rdd 
rO  —  Vc:  rdO  ::  r  :  rQ=  -77-7  x  r ;   consequently,   by   com- 

paring  this  value  of  rQ  with  that  of  (3)  at  p.  133,  we  must 

have  h'€  =  dr ;  and  of  course  if  dd  represents  the  differential 

of  6,  taken  for  the  independent  variable,  J/g  must  represent 

dr^  the  differential  of  7* ;  r  being  a  function  of  0,  see  (4.)  at 

p.  2.     We  also  have,  from  the  triangles  Sch^  and  S/'N,  the 

dr 
proportion  So  \Vc'.'.  S/'  :  7'N,  or  rdQ  :  dr  ::  r  :  ?'N  =  -^, 

which  is  the  same  as  (4)  at  p.  133. 

It  is  hence  evident,  that  if  the  angle  dd  is  infinitely  small, 
the  triangle  Sch'  and  the  curve  line  triangle  Sab  will  come 
infinitely  near  to  coincidence;  consequently,  according  to 
the  method  of  limiting  ratios,  we  shall  ultimately  have 

la  :  Sa  : :  Sr  :  rQ    or    dr  :  rdO  :  r  :  rQ  =  -j-^ 

dr 

for  tbe  subtangent,  and  rdO  :  dr  ::  r  :  'r'N  =  -j^  for  the  sub- 
normal; results  in  conformity  to  the  method  of  limiting 
ratios ;  see  p.  46. 

To  illustrate  what  has  been  done,  we  will  show  how  to 
draw  tangents  and  normals  to  the  common  parabola,  whose 
equation  is  4aa3  =  y^,  when  the  pole  is  taken  at  the  focus. 

Thus,  let  PAQ  represent  the  parabola,  having  AM  for  its 


ILLUSTRATIONS,    ETC. 


135 


axis  and  A  for  its  vertex ;  tlien  S,  being  tlie  focus,  we  liave 
4a  =  4AS,  and  representing  tlie  perpendiculars  AM  and 
PM  by  X  and  y,  the  equation  4AS  x  AM  =:  PM'  of  the 
parabola,  becomes  4aa?  =  y^ ;  which  expresses  the  equation 
referred  to  rectangular  co-ordinates.  Supposing  the  right 
line  T^  touches  the  parabola  at  P  and  intersects  the  axis  in 
T,  we  have  SP  =  r  and  the  angle  PST  included  by  PS  and 
TS  =  0.     By  trigonometry,  we  have 

PS  cos  ang  PSM  —  —  r  cos  0  —  SM  —  x  —  a, 
or     x  =  a  —  7'  cos  0,     and     PS  x  sin  PSM  —  rsin.d=zy  ; 

consequently,  from  the  substitution  of  these  values  in  the 
equation  4«a?  =  y-,  we  have  4a  (a  —  r  cos  0)  —  /^  sin"  0,  which 
is  easily  reduced  to  the  form 

4(2^  —  4:ar  cos  0  -f  r-  cos^  6^  {2a  — r  cos  6)^ 
=  T^  sin^  0  +  7^  cos^  Q  ^=1  r-   (since   sin^  0  +  cos-  Q  -z  V)\ 


consequently,  we  readily  get  r  =:  - — 


2a 


cos  0 


a 

cos-- 


for  the 


polar  equation.     By  taking  the  differentials  of  this  equation, 
--- ,  and  thence  -y-  =  cot  ^ ,  which, 


we  have  dr 


a  sm  -  dQ 


136  ILLUSTRATIONS,    ETC. 

substituted  for  (3)  in  p.  133,  gives 
/J 
r  cot  2  =  PS  X  tan  ang  SPT 

=  the  perpendicular  from  S  to  SP,  limited  by  the  tangent 
PT  =  the  sought  subtangent;  and  from  (4),  at  p.  133,  we 

.     d 

,         « sm  2  n 

have  ^7:  =  —  =  the  subnormal  =  r  tan  -  =  the 

cos«2 
perpendicular  to   SP  through    S,   produced    to    meet   the 
perpendicular  to  l^t  through  P,  which  gives  the  limit  of 
the  required  normal. 

Eemarks. — 1.  By  taking  the  differentials  of  the  members 
of  the  equation  4:ax  =  ?/^,  we  have  ^adx  =  2ydy^  which  gives 

— -  =  -^  and  thence  yx~=z-f-=z2x  =  the  subtansrent, 
di/      4a^  ^      dy       4:a  ° 

agreeably  to  what  is  shown  at  page  127  ;  consequently,  by 

taking  MT  =  2AM  =  2,r,  and  joining  P  and  T  by  a  right 

line,  it  will  touch  the  parabola  at  P. 

2.   From  4,adx  =  2ydy,  we  get  2a  =  ^~  =  subnormal 

(see  p.  128),  is  constant,  and  equal  to  -^  =  half  the  parameter 

of  the  axis  of  the  parabola ;  since  4a  is  called  the  parameter 
or  latits  rectum  of  the  axis  of  the  parabola. 

For  another  example,  we  will  show  how  to  draw  the  tan- 
gent and  normal  to  the  logarithmic  spiral,  whose  equation  is 
r=:a^;  by  using  polar  co-ordinates. 

Let  1,  2,  3,  Y,  represent  the  spiral,  having  r  for  its  pole, 
and  r,  1,  T  for  its  angular  axis,  such  that  the  positive  values 
of  0  are  the  arcs  of  a  circle  (rad.  =  1),  which  increase  arith- 
metically in  the  order  1,  2,  3,  Y,  while  ?'  =  a^  increases  geo- 


ILLUSTRATIONS,   ETC. 


137 


metrically ;  then,  because  0  =  0  at  1,  it  is  clear  that  since 
r  —  a^  =:  aP  =  Ij  that  we  must  have  r  =  1  represented  by 
rl,  and  in  r^=^a\  6  —  1  must  be  represented  by  the  arc  of 
the  circle  whose  length  is  1,  which  we  may  suppose  to  equal 
the  length  of  its  radius. 

By  taking  the  hyperbolic  logarithms  of  the  members  of 
r  =  a^,  we  have  log  r  =  6  log  a ;  whose  differentials  give 

—p  =  tan  ang  ?^ST  =  log  a;  conse- 
quently, the  angles  at  which  the  radius  vector  cuts  the 
spiral  having  the  constant,  log  a,  for  its  tangent,  must  be 
constant  or  invariable  ;  noticing,  if  log  a  =  1,  that  the  radius 
vector  cuts  the  spiral  at  an  angle  of  45°  or  half  a  right  angle. 

By  (3)  and  (4),  given  at  page  133,  if  we   divide  r  by 
—J-  =  log  a,  and  multiply  r  by  --,-  =  log  a,  we  shall  have 

and  r  log  a  =  r'N,  for  the  subtangent  and  sub- 


dr       -,  ,^ 

—  =z  loo:  a  ad,    or 


ra, 


log  a 

normal,  as  required ;  consequently,  the  tangent  and  normal 

can  readily  be  drawn. 

(6.)  When  a  right  line  touches  a  curve  at  an  infinitely 


138  ASYMPTOTES  DllAWN,   ETC. 

remote  point  from  the  origin  of  tlie  co-ordinates,  and  at  the 
same  time  passes  at  a  finite  distance  from  the  origin  of  the 
co-ordinates,  it  is  said  to  be  a  rectilinear  asynijptote  to  the 
curve. 

Thus,  by  assuming  the  equation  of  the  tangent  from  (1), 

given  at  page  129,  we  have  Y  —  y  =  -j^  {X  —  x),  in  which 

X  and  ?/  belong  to  the  point  of  contact  of  the  tangent  with 
the  curve,  while  X  and  Y  are  the  co-ordinates  of  any  other 
point  of  the  tangent ;  then,  by  putting  Y  and  X  successively 
equal  to  naught,  we  have 

-,=  J(X-.),     or     X^.-'^....(5), 
and  Y  =  2/-a,| (6); 

from  which,  it  clearly  results,  if  X  and  Y,  or  either  of  them, 
is  finite  when  a?  or  y  is  infinite,  that  the  curve  must  have  a 
rectilineal  asymptote,  while  if  X  and  Y  are  both  infinite  or 
impossible,  the  curve  has  no  rectilineal  asymptote. 


Thus,  from  y  =  -  Vx^—  «^,  the  equation  of  the  hyperbola, 

(Xi 


dx 

Vx""  —  rr 

dy- 

=  X- 

-  X 

a 

x'-a? 

we  have        ay  =^-  xdx  h-  \/{x^—  a^)     or 


which  witli  the  value  of  y  reduce  (5)  to  X 

which,  by  making  x  infinite,  and  rejectmg  a^  on  account  of 
its  minuteness,  with  reference  to  ar^,  becomes 

X  =  a? =  X  —  X  —  0; 

X 

consequently,  the  hyperbola  has  an  asymptote  passing  through 
the  center. 


ASYMPTOTES  DRAWN,    ETC.  139 

Again,  reducing  the  equation  to  i»  —  j  V(f^'^'^y%  ^^  have 
~  z=-  — ^,    and  thence   (2)   is   readily  reduced   to 

Y  =^  y —  ;  consequently,  making  y  infinite  and  re- 
jecting V^  on  account  of  its  comparative  smallness,  we  have 

Y=^y  —  -^=y  —  y=^0,  and  of  course,  as  before,  the 
curve  has  an  asymptote  passing  through  the  center. 

Eesuming  the  equation  y  ==  -  Vx^  —  a^,  and  supposing  x 
unlimitedly  great,  by  rejecting  a^  on  account  of  its  compara- 
tive smallness,  we  have  7  =  —  a? ;  or,  since  the  radical  oue-ht 

to  have  the  ambiguous  sign  ±,  we  get  2/  =  ±  -  a?,  or  its 

UlC  old 

equivalents,  3/  =  —    and  ?/  = ^  ;  which  clearly  are  the 

equations  of  two  right  lines  that  are  asymptotes  to  the 
hyperbola,  passing  through  the  center  of  the  curve,  in  ac- 
cordance with  what  has  been  before  shown;  noticing,  that 
equivalent  conclusions  result  immediately  from 

%^^^_j^ ^^^   dy  ^  h  V¥Tf^ 

dx  a  \/{x^  —  a?)  dx      a        y 

making  x  infinite  in  the  first  and  y  in  the  second,  and  reject- 
ing a?  in  comparison  to  a?^,  and  U^  in  comparison  to  y"^ ;  and 

we  thus  get  dy  =  ±  -  dx,  or  its  equivalents  dy  =  -  dx 

and    c?y  = d.x,  which  are  clearly  the  differentials  of  the 

equations  given  above. 


140  ASYMPTOTES  DRAWN,   ETC. 

Kemarks. — 1.   If  we  convert 

into  a  series  arranged  according  to  the  desc3nding  powers 
of  Xj  we  shall  have 

Ix  ( .       a'x--\        a^xr^       a^xr^         „ 

which,  when  x  is  very  great,  clearly  gives 

hx  ,    hx  /.       a}x~'^\ 

for  a  succession  of  lines  that  are  clearly  asymptotes  to  each 
other,  and  to  the  hyperbola;  noticing,  that  these  are  some- 
times called  hyperholie  asymptotes^  because  the  first  of  them 
are  right  lines.     It  is  evident  from 


that 

we  may, express 

the  asymptotes  in  terms  of  the  descend- 

mg  powers 

of  7J. 

2. 

From  ay  =  x^ 

-¥-- 

=  xH1l- 

-¥x^\ 

we  in  like  manner 

get 

y  = 

4('- 

"2 

¥x-^ 
8 

16 

12                       x 

,  &c. 

which  gives 

y  =  ± 

a' 

I/=± 

x}f 

¥x-\ 

a  V 

2  r 

for  asymptotes  to  the  curve  whose  equation  is  ay  =  x*  —  h\ 
and  to  each  other ;  and  because  none  of  these  are  rectilineal, 
they  are  from  their  forms  said  to  be  parabolic  asymptotes. 


ILLUSTRATIONS,   ETC.  141 

3.  It  is  hence  easy  to  perceive  how  we  may  proceed  to 
find  the  asymptotes  of  curves  that  admit  of  them.  Tlius,  to 
find  asymptotes  of  the  curve  whose  equation  is 

2/'V  —  jpx^  —  l)x  -^  c  ■=■  ^. 

Dividing  its   terms   by  a?^,    the   equation  is   reduced   to 

'if'  -  ■  jpx  —  l)x~^  +  cx~^  =  0  ; 

consequently,  if  —  px  =  0,  and  y^  —  px  —  hx~^  —  0, 
are  the  successive  parabohc  asymptotes  of  each  other  and 
the  proposed  curve. 

If  we  take  the  curve  whose  equation  is  "  mf  —  x^y  =  m^V' 
and  develop  y  into  a  series  of  the  descending  powers  of  ,t, 
we  shall  in  like  manner  get  y  =  ~  7n,  y  =  —  tti  —  in'^x^^^ 
y  z:^  —in  —  tn^x~^  —  ^mx-^^  &c.,  for  the  hyperbolic  asymp- 
totes of  the  proposed  curve ;  noticing,  that  the  first  of  these 
is  a  right  line  parallel  to  the  axis  of  x  on  the  side  of  y  nega- 
tive, drawn  at  the  distance  m,  from  x. 

4.  When  a  curve  is  referred  to  polar  co-ordinates,  it  is 
clear  that  there  will  always  be  an  asymptote  when  7',  the 
radius  vector,  is  infinite,  and  the  corresponding  value  of  Q  is 
finite ;  but  if  r  and  Q  are  both  infinite,  there  is  no  asymptote. 

(7.)  To  illustrate  what  has  been  done  more  fully,  take  the 
following 

EXAMPLES. 

1.  To  draw  a  tangent  and  normal  to  any  point  of  the 
logarithmic  curve,  and  to  determine  its  asymptote. 

Let  OACB  be  the  logarithmic  curve,  having  0  for  the 
origin  of  its  rectangular  co-ordinates,  and  OA,  BO,  &c.,  for 
its  ordinates  on  the  side  of  x  positive,  and  B'C,  D'E',  &c., 
on  the  side  of  x  negative ;  then,  y  =  a^,  representing  the 
equation  of  the  curve,  by  taking  the  hyperbolic  logarithms 
of  its  members,  we  shall  have  log  y  —  x  log  a,  so  that  x, 


142  ILLUSTRATIONS,   ETC. 


D'^      W  0  B  +X 


being  supposed  to  commence  at  0,  we  shall  have  OA  =  1 
for  the  unit  of  length,  and  log  BC  =  OB  x  the  hjrperbolic 
logarithm  of  a,  and,  changing  the  sign  of  a?,  we  have 
log  B'C  =  —  OB'  X  log  a,  and  so  on.  By  taldng  the  differ- 
entials of  the  members  of  the  equation  log  y  =  x  log  a,  we 

have  —  =  dx  log  a,  which  eives  ^7—  =  , =  7/1  =  the 

y  dy       loga 

ydy 
subtangent  =  const,  and  ^^-~  =  y^  log  a,  the  subnormal  [see 

(1)  and  (2)],  which  is  clearly  correct,  since  y  is  a  mean  pro- 
portional between  the  subtangent  and  subnormal;  hence, 
joining  the  point  of  contact  of  the  tangent  with  the  extremi- 
ties of  the  subtangent  and  subnormal,  the  tangent  and  normal 
required  become  known. 

To  find  the  asymptote :  by  changing  the  sign  of  a?,  the 

equation  y  =  a^  becomes  2/  =  a^  =  ~ ;  consequently,  since  a 

is  supposed  to  be  positive  and  sensibly  greater  than  unity,  it 

clearly  foUows,  from  y=z—,  that  if  x  is  unlimitedly  great,  y 
ct 

is  unlimitedly  small,  and  thence  the  axis  of  x  is  plainly  an 

asymptote  to  the  curve. 

Eemark. — It  is  evident,  from  what  is  here  done,  and  from 
what  has  been  done  at  p.  136,  that  the  logarithmic  curve  and 


r 


k 


ILLUSTRATIONS,   ETC.  143 

the  logaritlimic  spiral,  liave  resulted  from  different  ways  of 
expressing  tlie  relation  of  a  system  of  numbers  and  tlieir 
logarithms  by  linear  description. 

2.  To  draw  a  tangent  to  the  curve  whose  equation  is 
xy  —  A",  and  find  its  asymptotes. 

By  taking  the  differential  of  the  members  of  the  equa- 
tion, since  A-  =  const,  we  have  ydx  +  xdy  =.  0,  which  gives 

^- — 1-  a?  =  0     or    ^-r-  =  —  a?  =   the   subtano^ent ;   which, 
dy  dy 

being  negative,  shows  that  it  lies  in  a  contrary  direction 

from  what  has  heretofore  been  supposed,  or  that  it  falls  in 

the  direction  of  the  positive  values  of  x.     If  the  extremity 

of  the  subtangent  is  joined  with  the  point  of  contact  of  the 

tangent  and  curve,  the  tangent  will  of  course  be  drawn  as 

required.     Because  the  proposed  equation  is  equivalent  to 

A^  A^     .     . 

either  of  the  forms  y  =  —     or    a?  =  —  ;   it  is   clear,  from 
^      X  y 

the  first  form,  that  making  x  infinitely  great,  reduces  y  to 

an  infinitesimal ;  and,  from  the  second  form,  it  results  that 

indefinitely  great  values  of  y  give  infinitesimal  values  of  x ; 

consequently,  the   axes  of  x  and  y  are  asymptotes  of  the 

curve. 

Remark. — It  is  easy  to  perceive,  that  the  equation  xy=^A? 
is  that  of  the  hyperbola,  when  referred  to  its  asymptotes  as 
axes  of  co-ordinates. 

3.  To  find  the  subtangent  in  the  hyperbolic  spiral,  whose 
equation  is  rO  =  a. 

By  taking  the  differentials,  as  in  the  preceding  example, 
we  have  rdd  +  Odi'  =  0  ;  which  gives 

rdd  tHB 

-T-  ~  —0,     and     -T—  =  -~rO=  —a=  the  subtangent 


144  ILLUSTRATIONS,   ETC. 

[see  (3)  at  p.  133]  ;  consequently,  the  subtangent  is  negative 
and  equal  to  a.  It  is  hence  easy  to  perceive  how  the  curve 
may  be  constructed,  and  its  asymptote  drawn,  since  it  mani- 
festly has  an  asymptote. 


Thus,  describe  the  circle  ACBD  with  the  unit  of  distance 
as  radius,  and  draw  the  perpendicular  diameters  AB  and 
CD,  taking  the  first  of  them  for  the  angular  axis  and  the 
center  P  of  the  circle  for  the  pole  of  the  spiral ;  then,  pro- 
ducing PC  to  E,  such  that  PE  =  a^  and  drawing  EF  perpen- 
dicular to  it,  EF  will  clearly  coincide  in  direction  with  the 
asymptote,  on  the  supposition  that  the  values  of  0  in  the 
equation  rO  z=:a  are  estimated  from  A,  in  the  order  of  the 
letters  ACBD,  to  include  any  number  of  revolutions  that 
may  be  desired. 

To  describe  the  spiral  by  points ;  we  put  its  equation  in 

the  form  r=^-^  and  thence,  since  the  semicircumference  of 
u 

the  circle   ACB  ==  tt  =  3.1416   very  nearly,  from  knowing 

a,  we  readily  find  the  corresponding  values  of  r. 

Thus,  by  taking  the  arc  Al  equal  to  the  radius  of  the 

circle  =  unity,    by   drawing   a   line   from  P  through  1  to 

equal  a,  we  have  a  point  in  the  spiral ;  and  setting  the  arc 

Al    from    1    to    2,    and  making   a   line   from   P   through 

2  =  X,  we  have  another  point  in  the  spiral;   and  in  like 


ILLUSTRATIONS,    ETC.  145 

manner,  by  setting  tlie  arc  1,  2  from  2  to  3,  and  drawing  a 
line  from  P  througli  3  =  ^ ,  we  have  another  point  in  the 

o 

spiral ;  and  so  on,  indefinitely.  Hence,  drawing  a  curve 
with  a  steady  hand  through  the  points  found,  we  shall  have 
•an  approximate  representation  of  the  spiral,  which  evidently 
has  EF  for  its  asymptote. 

K  EM  ARK. — It  is  manifest  that  this  curve  took  its  name 
from  the  striking  analogy  between  its  equation  rO  =  a,  and 
that  of  the  hyperbola  xy  ■=^  a^\  see  page  119  of  Young's 
"Differential  Calculus." 

4.  To  find  the  subtangent  in  the  spiral  of  Archimedes, 
whose  equation  is  r  —  a0. 

By  taking  the  differentials  of  the  members  of  its  equation, 

dv 
we  have  —=:«.==  its   subnormal  =  const,  and  of  course 

T^'  ■—  -T-^^  r  Y.  -y—  zzzT^  -■=  rO ;  consequently,  the  subtangent 

equals  the  length  of  a  circular  arc  radius  r,  and  angle  that 
between  r  and  the  angular  axis ;  see  Young,  page  118. 

Eemark. — The  equations  of  this  and  the  hyperbolic  spiral 
are  included  in  the  class  of  spirals  represented  by  the  equa- 
tion r  =  ad'^ ;  noticing,  that  n  may  be  positive  or  negative, 
according  to  the  nature  of  the  case. 

5.  To  find  the  subnormal  and  subtangent  in  the  spiral, 
whose  equation  is  {r^  —  or)  6^  =  J^. 

Solving  the  equation  with  reference  to  r,  we  have 

A"  4) 

which  gives 

dr  ¥  l^ 


'^'  Va^  +  4  ^Va^^-^  +  J^ 


=  the  subnormal. 


146  ILLUSTRATIONS,  ETC. 

Dividing  'r  by  the  subnormal,  we  "have  —r-  =  —  ^^ jr^ 

for  tbe  subtangent^  which  reduces  to  —h  when  6  =  0. 

Remarks.— 1.  From  r  =  yW  +  -^),  it  is  evident  that 
the  last  value  of  r  is  a ;  which  immediately  follows  from  the 
proposed  equation,  when  put  under  the  form  (P  =  -^ s . 

Hence,  if  from  the  pole  of  the  spiral  as  center  with  a  as 
a  radius,  a  circle  is  described,  it  will  clearly  be  an  asymptote 
to  the  spiral ;  since,  when  r  =  a^  0  must  be  unlimitedly 
great,  or  must  include  an  unlimitedly  great  number  of  cir- 
cumferences. 

2.  In  much  the  same  way  it  may  be  shown  that  the  spiral 

whose  equation  is  r  =  y  la^  —  -^1,  lies  wholly  within  the 

circumference  of  the  circle  whose  radius  is  a ;  the  circum- 
ference being  an  asymptote  to  the  spiral. 

6.  To  find  the  subnormal  and  subtangent  of  the  spiral 
whose  equation  is  (/^  —  ar)  ^  =  1. 

From  the  equation  we  readily  get  7^  —  ar  =  -^^,  and  thence 


/•  =  I  +  Y^j  +  ^),  which  gives 


-^  =  — *Ko  -^V  T-  +h= y— =  "t^®  subnormal ; 

consequently,  dividing  r*  by  this,  we  readily  get  the  sub- 
tangent. 

Remarks. — From  r  =  -^  +  y\^  '^  ey'  ^^  ^^^  proposed 

equation,  it  follows  that  the  circumference   of  the  circle 
whose  radius  =  a,  is  an  asymptot-e  to  the  spiral,  being  an 


ILLUSTJIATIONS,   ETC.  147 

interior  asymptote ;  while  tlie  circle  whose  radius  is  a,  is  an 
exterior  asymptote  to  the  spiral  whose  equation  is 

{ar-r')6'  =  l,     or     ,.  =..  |  +  |/(^  -  1), 

the   spiral   falling  wholly  within  the   circle:   see  Young's 
"  Differential  Calculus,"  p.  123. 

7.  To  find  the  subnormal  and  subtangent  of  the  paraboUo 
spiral^  whose  equation  is  r^  =:  a-0    or    7'  --  a0  . 

By  taking  the  differentials,  we  have 

dr      1  a  a  .i         i  i 

consequently,  7*^  divided  by  the  subnormal,  gives 

~d^ 


-j^  =  2a0^  =  the  subtangent, 


and  since,   from  the  proposed   equation  0*  =  - ,  we  have 
r'dO  _  2/'^ 

Eemaek. — It  is  manifest  that  the  spiral  is  called  parabolic, 
from  the  analogy  of  its  equation  r- z=  o/d  to  that  of  tbe 
parabola  y"  =  ax. 

8.  To  find  the  subtangent  and  subnormal  at  any  point  of 
the  cissoid  of  Diodes. 


■i      Let  ABCD  be  a  semicircle,  baving  AD  for  its  diameter,  to 


148  ILLUSTRATIONS,  ETC. 

whicli  tlie  perpendicular  ordinates  BF  and  OE  are  drawn,  at 
equal  distances  OF  and  OE  from  the  center ;  then,  drawing 
a  right  line  from  A  to  C,  the  extremity  of  the  ordinate  OE, 
it  will  intersect  the  other  ordinate  at  a  point  Q-  of  the 
cissoid. 

Kepresenting  AF,  FGr,  and  AD,  severally,  by  a?,  y,  and  a ; 

the  equiangular  triangles  AFGr  and  AEG  will  (from  known 

principles  of  geometry)  give  the  equal  ratios  expressed  by 

y      CE  .  .         BF 

-  =  -:-^  =  (by  construction  and  the  nature  of  the  circle)  zf^^ , 
a;      AE      ^  -^  ^  DF' 

y"      BF^      AFxDF      AF         ^       ^    ^^       . 

or    ^  =  YSTTTJ  =  — TTf^5 —  =  TTPT  = by  the  nature  of 

x"      DF^  DF^  DF      a  —  x     ^ 

the  circle ;  consequently,  we  shall  have  if-  — ,  for  the 

(.1  —  a? 

equation  of  the  cissoid.  By  taking  the  differentials  of  the 
members  of  this  equation,  we  have 

,  .  ,     .  ydy      a?  (Sa  —  2,??)        ,        ,  , 

"which  gives      ^-j^  =  —^— — — ^  =  the  subnormal, 
dx  £1  \Oj  —  x\ 

and  dividing  y^  by  the  subnormal,  we  have 

2a7  {a  —  X) 


3a  —  2a; 

r3 


the  subtangent. 


If  in  y-  = we  put  x  —  a,  we  shall  have  if  — =  ^ » 

consequently,  if  is  infinitely  great,  and  of  course  there  must 
be  two  infinite  values  of  y,  one  of  which  is  expressed  by 
+  y  and  the  other  by  —  ?/ ;  which  must  be  asymptotes  to 
the  cissoid. 

Thus,  from  ?/-  =: we  have  ?/  =  +  a?  i/ ;   con- 


149 
sequently,    any  positive    value    of  y  being   expressed   by 


y 


=  xi/ ,  the  corresponding  negative  value  must  be 


^    a  —  X 


FG 


of  course,   tlie 


expressed  by  3/  = 

lower  part  of  tbe  curve  must  be  identical  with  tbe  upper, 
being  described  in  the  lower  semicircle,  after  the  method 
that  has  been  used  in  describing  AGL.  It  is  hence  manifest 
that  a  perpendicular  to  the  diameter  AD  through  D,  when 
produced  infinitely  both  w^ays,  will  represent  the  asymptotes 
of  the  branches  of  the  curve. 

Eemaek. — It  is  easy  to  perceive,  that  the  upper  and  lower 
branches  of  the  curve  will  touch  each  other  at  A,  and  will 
form  with  each  other  what  is  called  a  cusp  of  the  first  hlnd^ 
since  their  convexities  touch  each  other.  It  may  be  added, 
that  if  two  branches  of  a  curve  touch  each  other  in  such 
a  way  that  the  convexity  of  one  is  in  contact  with  the 
concavity  of  the  other;  then  they  form,  at  their  point  of 
contact,  what  is  called  a  cusp  of  the  second  kind. 

9.  To  draw  a  tangent  and  normal  at  any  point  of  the 
common  cycloid. 


Let  BFD  represent  the  circumference  of  a  circle  having 
0  for  its  center,  OB  =  r  for  its  radius,  DE  and  EF  for  the 


150 

rectangular  co-ordinates  of  the  extremity  F  of  the  arc  DF, 
the  point  D  of  the  extremity  of  the  diameter  DB  being 
taken  for  the  origin  of  the  co-ordinates ;  then,  if  the  ordinate 
EF  in  the  circle  is  produced  to  G  so  as  to  malce  FGr  =  the 
circular  arc  DF,  G  will  be  a  point  in  the  cycloid.  Hence, 
representing  DE  by  a?,  and  EG  by  2/,  we  shall  have 

y  =  EF  +  the  arc  DE, 

or  since  DE  =x  =  the  versed  sine  of  the  arc  DF,  and  EF  = 
the  sine  of  the  same  arc,  when  r,  the  radius  of  the  circle,  is 
taken  for  the  radius;  then,  denoting  the  arc  (according  to 
usage)  by  ver  sin-^a?,  we  shall  have 

y  =z  ver  sin~  ^  a?  +  sin  ver  sin~  ^  x 

for  the  equation  of  the  cycloid,  when  the  origin  of  the 
co-ordinates  is  taken  at  D,  called  if  Ad?  vertex  of  the  curve.  By 
taking  the  differentials  of  the  members  of  the  equation,  we 
shall  have 

,      7'dx  rdx  —  xdx       .  /2r  —  x   , 

dy  =  -   H =:=zirrrrr=:  =  Y    dx  ] 

V2rx  —  x^       V2rx  —  x^  -^ 

since   (see   page   73)  -—7:- — '■ ^r  is  the  differential  of  the 

4/(2/'a?  —  xr) 

arc  whose  versed  sine  is  x  and  radius  r,  and  that      -    — r=z 

V2rx  -  x^ 

is  the  differential  of  the  sine  of  the  same  arc.     Hence, 

dx         /     X  X  DE 


V^27^ 


dy       y  2r-x       V27-x-x^      EF  ' 
consequently,  since  -j-  equals  the  tangent  of  the  angle  which 

the  tangent  to  the  cycloid  at  G  makes  with  EG,  and  that 

DE 

^^r  equals  the  tangent  of  the  angle  which  the  chord  of  the 

arc  DF  makes  with  EF,  it  results  that  the  tangent  to  the 


ILLUSTRATIONS,    ETC.  lol 

cycloid  at  Gr  is  parallel  to  the  chord  DF  of  the  corresponding 
arc  DF  of  the  circle.  Hence  a  right  line  drawn  through  Gr 
parallel  to  the  chord  DF  will  be  a  tangent  to  the  cycloid  at 
G ;  and  because  the  chords  of  the  arcs  DF  and  BF  cut  each 
other  perpendicularly,  it  follows  that  a  right  line  drawn 
through  Gr  parallel  to  the  chord  BF,  and  extended  to  meet 
DB  produced  toward  B,  will  be  a  normal  to  the  curve 
at  G. 

Eemarks. — 1.  If  in  ^—  =  the  subtangent,  we  put  for  y 
its  value,  we  shall  have  the  subtangent  = 


/ 


TT X  (ver  sin~^  a?  -[-  sin  ver  sin~^  a?)  ; 

consequently,  having  computed  the  value  of  this,  and  set  it 
off  from  E  on  BD  produced  toward  D,  by  joining  the 
extremity  of  the  produced  part  with  Gr,  we  shall  clearly 
have  the  tangent,  as  derived  from  the  subtangent  and  the 
point  of  contact. 

2.  Admitting  the  construction  of  the  figure,  and  that  KL, 
the  diameter  of  the  semicircle  KIL  perpendicular  to  AB,  is 
equal  to  DB ;  we  shall  have  AB  =:  BC  =  the  semicircum- 
ference  DFB  or  KIL,  arc  KI  ==  arc  DF  =  FG  =  HE  =  LB 
(since  EF  =  HI),  and  of  course  arc  IL  ==  AL.  Hence,  we 
shall  have  AM  =  AL  —  ML  =  arc  IL  —  its  sine  IH ;  conse- 
quently, if  y  represents  AM,  and  IM  =  HL  =  ver  sin  arc  IL, 
we  shall  have  y  =  ver  sin~^i»  —  sin  ver  sin~^a?  for  another 
form  of  the  equation  of  the  cycloid ;  which  may  be  regarded 
as  being  a  transformation  of  the  equation  previously  found, 
when  the  origin  of  the  co-ordinates  is  changed  without 
changing  their  directions. 

3.  The  preceding  equation  clearly  suggests  the  ordinary 


152 

method  of  describing  the  cycloid.  Thas,  conceiving  the 
circle  whose  diameter  is  KL,  to  have  the  point  I  placed  at 
A,  and  then  rolled  (without  any  sliding)  from  A  toward  C, 
the  point  I,  in  one  revolution  of  the  circle,  will  manifestly 
describe  the  cycloidal  arc,  ADC ;  noticing,  that  AC  and  BD 
are  called  the  base  and  axis  of  the  curve,  and  that  the  circle 
described  on  the  axis  is  called  the  generating  circle.  It 
may  be  added,  since  x  and  Virx  —  x^  are  (from  the  princi- 
ples of  trigonometry),  not  only  the  versed  sine  and  sine  of 
the  arc  IL,  but  of  the  arc  IL  increased  or  diminished  by  any 
number  of  times  the  circumference  of  the  circle  whose  diam- 
eter is  KL,  we  may  suppose  the  circle  to  roll  on  infinitely 
along  the  right  line  AC  produced  to  infinity  toward  C,  and 
thereby  to  describe  an  unlimited  number  of  successive  cy- 
cloids, which  will  all  be  comprehended  in  the  preceding 
equation. 

10.  To  draw  a  tangent  to  the  curve  whose  equation  is 
y  =  Zx-\-Ux'-  2ar\ 


A  B  .  D 


By  taking  the  differentials  of  the  members  of  the  equa- 
tion,   we  have   -j-  =  3  -f  36a?  —  6u?-;    consequently,   the 

equation  of  the  tangent  (see  p.  126)  Y  —  ?/  =  -^  (X  —  a?),  is 

easily  found  for  any  proposed  value  of  x. 
Thus,  if  we  put  1  for  a?,  we  have 

^  =  3  +  36  -  6  =  33,     and    ^^  =  3  +  18  -  2  ==  19  ; 


ILLUSTRATIONS,   ETC.  153 

consequently,  tlie  equation  of  the  tangent  is 

Y  -  19  =  83  (X  -  1),     or    Y  =:  33X  -  14. 

It  is  easy  to  perceive  that  this  tangent  cuts  the  curve ; 
since,  by  putting  x  and  y  for  X  and  Y,  it  is  immediately 
reduced  to  x"  —  ^j?  +  15a.'  —  7  =  0,  whose  roots  are  a?  =  1, 
a?  =  1,  and  a?  ==  7.  The  first  two  of  these  roots  belong  to 
the  point  of  contact,  while  a?  =  7  is  a  point  at  which  the 
tangent  cuts  the  curve,  having  y  =  257  for  the  correspond- 
ing ordinate  of  the  curve. 

If  we  put  X  =  4:  in  -^  =  3  +  8 6a?  —  6x^,  and  proceed  in 

ClX 

the  same  way  as  before,  we  shall  get  -~-  =  51,  and  thence 

the  equation  of  the  tangent  to  the  curve  at  the  point  whose 
abscissa  is  4,  is  Y  —  172  =  51  (X  —  4).  Putting  x  and  y 
for  X  and  Y  in  this,  we  readily  get  x^—  9x'  +  24aj  —  16  =  0; 
whose  roots  are  a;  =  4,  x  =  4,  and  x  =  l,  the  first  two  of 
which  are  the  same  as  the  abscissa  of  the  proposed  point ; 
consequently,  the  tangent  cuts  the  curve  at  the  point  whose 
abscissa  =  1,  and  whose  corresponding  ordinate  is  y  =:  19. 

Because  the  tangent  to  the  curve  at  the  point  whose 
abscissa  is  1  cuts  the  curve  at  the  point  whose  abscissa  is  7, 
v/hile  the  tangent,  to  the  curve  at  the  point  whose  abscissa 
is  4  cuts  the  curve  at  a  point  whose  abscissa  is  1,  it  is  mani- 
fest that  the  first  of  these  tangents  must  touch  the  convex 
part  of  the  curve ;  that  is,  that  part  which  is  convex  toward 
the  axis  of  x ;  while  the  second  tangent  touches  that  part 
of  the  curve  which  is  concave  to  the  axis  of  x. 

It  is  hence  evident  that  there  must  be  a  point  in  the  curve 
whose  abscissa  is  between  1  and  4,  such  that  the  tangent  to 
the  curve  will  not  cut  the  curve  at  any  other  point.  Thus, 
1* 


154  ILLUSTRATIONS,   ETC. 

the  tangent  to  the  curve  at  the  point  whose  abscissa  is  3,  by 

putting  3  for  x,  reduces  ~  =  3  +  S6x  —  Gxr  to  -p  =  57, 

and  2/  =  3^  +  X'^j?  —  2jd^  becomes?/  =  117;  consequently, 
the  equation  of  the  tangent  becomes  Y  —  117  =  57  (X  —  3), 
or  Y  =  57X  —  54.     Hence,  putting  x  and  y  for  X  and  Y, 
we  have      Sx  +  ISar'  -  2ar»-  117  =  57  (a;  -  3), 
or,  3^  -f  ISr^  -  2x^  =  blx  -  54  , 

which  is  equivalent  to 

a?5-  9^  +  27^' -27  =  0; 

whose  roots  are  a?  =  3,  a?  =  3,  a?  =  3,  and,  of  course,  the 
tangent  to  the  curve  at  the  point  whose  abscissa  =  3,  cuts 
the  curve  at  the  same  point.  Because  the  curve  changes  the 
direction  of  its  curvature  at  C,  or  at  the  point  whose  abscissa 
is  3,  it  is  said  to  have  a  point  of  inflection  or  contrary 
flexure  at  C. 

Bemarks. — 1.  It  clearly  results  from  what  has  been  done, 
that  in  curves  which  suffer  an  inflection,  a  line  which  touches 
the  curve  on  one  side  of  a  point  of  inflection  may  cut  it  on 
the  other  side. 

2.  Because  the  tangents  cut  the  curve  at  their  point  of 
contact,  it  is  clear  that  the  points  of  contact  of  the  tangents 
may  be  regarded  as  the  union  or  coalescence  of  the  two 
points  in  which  the  curve  is  cut  by  a  secant,  by  regarding 
the  points  of  intersection  of  the  secant  as  being  unlimitedly 
near  each  othen  Also,  because  the  tangent  at  the  point  of 
inflection  does  not  cut  the  curve  at  any  other  point,  it  is 
clear  that  the  tangent  at  this  point  ought  to  be  regarded  as 
being  both  a  tangent  and  secant;  that  is,  as  cutting  the  curve 
and  as  tangent  to  its  convex  and  concave  arcs  at  the  point, 
when  taken  separately. 


RULE   FOR  THK   POINTS   OF   IN"FLECTI0:N".  155 

3.  If  the  point  of  contact  of  a  tangent  and  two  iinlimitedly 
near  points  of  intersection  of  a  secant  witli  a  curve  are  sup- 
posed to  be  equivalent,  it  is  easy  to  perceive  that  it  results 
from  what  has  been  done,  that  the  curve  may  be  cut  by  a 
right  line  in  three  points,  or  as  many  as  there  are  units  in 
the  degree  of  its  equation.  It  is  also  manifest  that  curves 
which  admit  of  a  point  of  inflection  must  be  at  least  of  the 

third  degree. 

cixi 

4.  Supposing  the  equation  of  the  tangent  Y— y=  -i^  (X— a?) 

Y  —  y        cly 

to  be  reduced  to  the  form  ^ — ^  =:  0 ;  then,  if  the 

X  — a?        dx  '  ' 

tangent  cuts  the  curve,  or  can  be  made  (as  in  the  question) 

to  cut  it  at  the  point  whose  co-ordinates  are  X  and  Y,  such 

Y  —  y 

that  Y —     ^ay  ^^  regarded  as   a  consecutive  value   of 

-^^  it  is  clear  that  for  the  preceding  equation  we  may  write 

d-y 

j^=z  0   or  infinity,  according  to  the  nature  of  the  case. 

See  page  13. 

Hence  the  points  of  inflection  of  a  plane  curve  may  be 
found  by  the  following 

RULE. 

1.  Let  X  and  y  represent  the  co-ordinates  of  a  point  of 

inflection,  and  suppose  -^^  =  F  (x)  =  a  function  of  x.    Then, 

proceed,  as   in  finding  maxima  and  minima,  to  find  the 
maxima  and  minima  of  F  (x) ;  that  is  to  say,  find  those  roots 

oi  — j~-'  =  0  which  do  not  reduce  —rV^  to  naught,  and 
cix  cLx 

dY  ix) 
they  will  correspond  to  points  of  inflection  ;   if  —}—  =  in- 


156  POINTS  OP  INFLECTION. 

dx 
finity,  we  must  find  the  roots  of   .^,  ^  =  0,  and  then,  as  m 
*''  dF{x)         '  ' 

maxima  and  minima,  find  those  which  correspond  to  maxima 

or  minima  which  will  give  points  of  inflection,  while,  if  there 

are  no  maxima  or  minima,  there  can  not  be  any  points  of 

inflection. 

If  the  roots  of  — ~-^  =  0  are  also  roots  of  — j~~  =  0, 

(PY  (x) 
we  must  find  the  roots  of  — j—  =  0,  provided  they  do  not 

reduce       ,  ^  ^  to  naught ;  and  so  on,  as  in  finding  maxima 

and  minima, 

2.  To  determine  for  any  value  of  a?,  whether  the  curve  is 
convex  or  concave  toward  the  axis  of  a?,  we  substitute  the 

value  of  a?  in  -T^  =  — ^  ;  then,  if  the  result  has  the  same 

sign  as  y,  it  is  easy  to  perceive  that  the  convexity  of  the 
curve  is  turned  toward  the  axis  of  x,  and  vice  versa.    Thus, 

from  ^  =  S  +  S6x  -  6.zr^,   we  have  ^^^  =  36  -  12^ ; 

which  is  clearly  positive  when  x  is  less  than  3,  and  the  con- 
vexity of  the  curve  is  turned  toward  the  axis  of  x ;  noticing, 

d¥  (x\ 
that  — ^  =  0   gives  36  —  12aj  =  0,    or  a?  =  3,  and  that 

— j-~  =  —  12  shows  F  (x)  to  be  a  maximum,  when  the 

curve  passes  from  being  convex  toward  the  axis,  to  being 
concave. 

To  illustrate  what  has  been  done,  take  the  following 

EXAMPLES. 

1.   To  find  the  point  of  inflection  in   the  curve   whose 
equation  iay  =  x^  +  a^. 


EXAMPLES.  157 

Here  we  liave  -^  =  Y  {x)  =  -^  x~^  +  2x!,   wliich.  gives 

d¥{x)  1    -I      o         1     ^'F(.^)        3  , 

__iJ  ^  _  _^  t  +  2     and     -^  =  p  ;    consequently, 

„  dF{x)  .  _  1      -I     ,     O  A  ^  A 

from  — :r~  =  0,  we  nave  —  t  ^   ^+2  =  0,    or    a?  =  t  5  ^'^^ 
dx  '  4  '  4' 

c^F  (;»)  .  .  .       '     ,  ^.  .    .  ,     ^  ^ 

Since  — Tx    ^^  positive,  ±  (a?)  is  a  minimum.     And  because 

— -T-^—  —  —  J  a?~^  +  2  lias  a  sign  contrary  to  tliat  of  y  wlien 
X  is  less  than  j ,  it  is  clear  that  its  concavity  before  a?  =  j 

is  turned  toward  tlie  axis  of  a?,  but  after  a?  =  j?  "^^^  ^^g^  ^^ 
"^  is  the  same  as  that  of  y  ;  consequently  the  curve  has 

an  inflection  at  the  point  whose  abscissa  =  -r . 

2.  To  find  the  point  of  inflection  in  the   curve  whose 
equation  \^y-  ^=^x  +  x". 

From  this  equation  we  have 

dx  -"^  y^')-      2y      ' 

,  ^,  dF  (x)       6x       dyl  +  Sx^       ^ 

and  thence      — x^-  =^ ^  —77-^—  =  0, 

dx  2y       dx     2y^ 


III/  X  -r  ^<^j  \  J.  1-  t)\jo  ^ 

or 


_  %  1  +  3a;^  _  (1  +  3a^y 
«;a?       y       "~2(r^+ar^)' 


which  readily  reduces  to  x^  +  2«^  =  - ,  whose  solution  gives 

3.  To  find  the  point  of  inflection  in  the  curve  whose 
equation  is  2/  =  a?^. 


158  POINTS  OF  INFLECTION. 

Here  -^  =  F  (x)  =  3a^,  gives 

— -,-^-^  =  ox    and     — r^-  =  6  ; 
ax  ax^ 

clF  (x) 
consequently,  from       ,    ^  =  0,  we  have   6.v  =  0,  or  a?  =  0, 

(PF(x) 
and  from      ,  ^  ^  =  6,  it  follows  that  a?  =  0  makes  F  {x)  =  Ss^ 
cCx^ 

SL  minimum. 

Because  -^  =  Sx^  equals  0  at  the  origin  of  the  co-ordi- 
nates, it  is  clear  that  the  curve  has  an  inflection  at  the  origin 
of  the  co-ordinates,  where  the  curve  touches  the  axis  of  a?  on 
the  side  of  x  positive  and  negative,  so  that  the  convexity  of 
the  curve  above  and  below  the  axis  of  x  is  turned  toward  it. 
Thus  the  curve  must  be  of  the  general  form  expressed  by 
the  adjoined  figure. 


Remark. — Any  curve  whose  equation  is  cf  the  form 
y  =  a?",  such  that  ?i  is  an  odd  positive  integer  greater  than 
one,  must  clearly  be  of  the  same  general  form  as  before. 

4.   To  find  the  point  of  inflection  of  the  curve  whose 

equation  is  y  =  a  -\-  (h  —  x)\ 

^^^^  ^  ""  ^^^  ""  ~  8  (^-^)%  gives  —^^  =  —  {h-x)  * 
or  '  =  "in  '  consequently,  putting  this  equal  to 
naught,  we  have  x  =  l)^  and  a  point  of  inflection  may,  of 
course,  exist  at  this  point      From  -^  =  —  -  {h—xy,  for 


EXAMPLES.  159 

x  =  h,  we  have  -~  =  0^  and,  of  course,  tlie  tangent  to  the 
curve  at  tlie  extremity  of  the  ordinate  y  =  a  is  parallel  to 

the  axis  of  x.    From  -t^^t— ,  =       ^,^—  it  is  clear  that  when 
dF  (x)  10 

(Ix 
X  is  less  than  5,  -tt^tt-t  will  be  positive,  and  when  x  is  greater 
ctij  \X) 

than  h,  -w^j--    will  be  negative;   consequently,   the   curve 

crosses  the  tangent  at  the  extremity  of  the  co-ordinate  a, 
where  it  has  a  point  of  inflection. 


0    "b 


Thus,  as  in  the  scheme,  the  curve  passes  through  the  ex- 
tremity of  the  ordinate  a,  the  point  of  inflection,  and  touches 
the  tangent  at  tlie  point,  above  and  below,  so  that  from  the 
nature  of  a  tangent  its  convexities  will  be  turned  toward 
the  tangent. 

5.  To  find  the  point  of  inflection  in  the  curve  whose 
equation  is  y  —  rnx  -\-  (h  —  x)^. 

As  in  the  preceding  example,  we  take  the  differentials, 

and  get  ¥  (x)  =  —-  =:  m  —  )r  (b  —  x)^ : 

^  ^        ax  3  ^  ^ 

hence,  as  before,        ^-^^=  Jq  (^  -  ^)\ 

and  thence  the  curve  has  a  point  of  inflection. 

Hence  clearly,  if  we  change  a  in  the  preceding  scheme 
into  7nh,  and  draw  the  tangent  at  its  extremity  to  make  an 


160  POINTS  OF  INFLECTION. 

angle  with  the  axis  of  a?,  whose  tangent  :=  m ;  then,  the 
curve,  when  drawn  with  reference- to  the  tangent,  as  in  the 
scheme,  will  express  the  curve  and  its  point  of  inflection,  as 
required. 

Remark. — For  the  curve  whose  equation  is 
y  z=  mx  +  {x  —  1))\ 

we  obtain  the  same  results  as  before;  with  the  exception, 
that  the  part  of  the  curve  toward  the  origin  of  the  co- 
ordinates lies  below  the  tangent,  while  the  remaining  part  .of 
it  is  above  the  tangent 

6.  If,  as  is  generally  the  case  with  spirals,  the  equation  is 
referred  to  polar  co-ordinates,  then  we  may  proceed  to  find 
its  points  of  inflection  as  follows. 


Thus,  let  WSY  represent  a  spiral  having  r  for  its  pole, 
T'T  for  its  angular  axis,  and  rS  for  its  radius  vector,  which 
makes  the  angle  0  with  rT;  then,  supposing  the  curve  to 
have  a  point  of  inflection  at  S,  it  is  manifest,  since  the 
tangent  to  the  curve  at  S  cuts  it,  and  touches  its  convex  and 
concave  arcs  at  the  same  point,  we  may  suppose  the  perpen- 
dicular from  the  pole  r  to  the  tangent  at  S  to  be  constant, 
when  a  small  change  is  made  in  the  position  of  the  point  of 
contact;  see  Vince's  "Fluxions,"  pp.  123  and  124. 

Supposing  with  Yince,  as  we  clearly  rnay  do,  that  the 

-  I  ,  by  taking 


EXAMPLES.  161 

the  differentials  of  its  members,  we  have 

dO  =  7)i  1-)        — ,     or    rdd  =  771  I -)     dr.  - 
\aJ         a  \a/ 

Hence,  for  simplicity,  referring  to  the  figure  at  p.  133,  we 
have  rdd  =  S<?    and    dr  =  h'c^  which  give,  since 

S¥=  V?~W+d?  =  |l  +  m^  (-)'"'|    dr. 

Hence,  drawing  the  perpendicular  p  from  the  pole  r^  to 
the  tangent  T^,  we  get  from  similar  triangles, 

^,,   ^       ^,  T^dd  '^  \a)    ''^'^  mr^^^ 

Sy :  Sc::  br:^  =  -^^t-  = -— = ; 

bo         r  /,^\2m]i  {(x'-^-\-rr^r'^^Y 


{1  +  m=g)'   I  dr 

which  agrees  with  the  perpendicular  Sy  found  by  Yince,  at 
p.  124. 

Because  of  the  supposed  constancy  of  ^,  the  differential 
of  this  must  be  put  equal  to  naught,  and  of  course  the 
differential  of  its  square  must  also  equal  naught;  conse- 
quently, we  shall  have 

(2W_+  2)  m^y^^m  +  l^^ %n^T^^  +1        _ 

or  2m^  /'^'^  + 1  +  (2m  -f  2)  m^  ^-^  ^sm  + 1  _  q^ 

1 

(m  +  1\'^"* 
2—1      X  a;   t'le    tarne 

conclusion  as  obtained  by  Yince.     To  make  r  positive  and 

real,  it  is  clearly  necessary  that  m  should  be  negative  and 

numerically  greater  than   1 ;   thus,  if  m  —  —  2,   we   have 

/1\"*  1 

^  =  (~^j      a  =  (4)^  X  a  =  ^2  X  a,  and  the  equation  of  the 


162  POINTS  OF  INFLECTION". 

spiral  d=l-j    becomes  6=  l-j      or-  =  0~*,   which   is 

equivalent  to    r  =  «0~%  the  equation  of  the  spiral  that  is 
called   the  lituus.     If  ?/i  =  —  3,   the   preceding   equation 

gives  r={^\    x  a,    and   the    equation    of   the    spiral    is 
or    r  =  ad~^ ;  and  in  like  manner  the  spirals 


©■ 


whose  equations  are?'  =  <20~*,  or  r  =  aO~^\  and  so  on, 
have  points  of  inflection;  while  all  those  spirals  in  which 
m  is  not  negative  and  numerically  greater  than  1,  have  no 
points  of  inflection. 


SECTION  VI. 

RADII   OF   CURVATURE,    INVOLUTES   AND   EVOLUTES,    ETC. 

(1.)  Supposing  (x  —  x'f  +  (?/  —  y'f  =  r  to  be  the  equa- 
tion of  a  circle,  whose  radius  is  r  and  the  rectangular 
co-ordinates  of  its  center  are  x'  and  \j' ;  then,  if  y  =  F  {x) 
represents  the  equation  of  any  plane  curve,  such  that  we 

can  find  ~   and  d  ~  ~  dx  from  it,  so  that  they  shall  be 
UjX  clx 

the  same  as  in  the  circle ;  then,  the  radius  of  the  circle  is 

called  the  radius  of  curvature  of  the  curve  at  the  point,' 

whose  co-ordinates  are  expressed  by  x  and  y.     Kepresenting 

-f-  hj  p  and  d  J-  -^  dx  =  -J-  by  ^'  in  the  proposed  curve, 

by  taking  the  first  and  second  differentials  of  the  equation 
of  the  circle,  on  the  supposition  of  the  constancy  of  r,  x, 
and   ?/',    we   shall  have   (y  —  y')  dy  -\-  (x  —  x')  dx  =  0 ; 

or,  since  -^  and  d  -^  ~  dx  must  equal  j?  and  p',  we  shall 

have  {y  ~  y')  jp  -{-  X  —  x'  =  0,  whose  differential  gives 
(y"~y')i^'  -f  j?2  +  1  =  0.  From  i^y—y')  p  +  x—  x'  —  0  and 
(^  _  ^J  +  (2^  _  yj  ^  r'  we  get  {y  -  yj  (/-  +  1)  =.  r^  and 
from  {y-y')p'=—  (/-  +  !)  we  have  (y— y)V'=  (i^'+ 0' i 
consequently,  from  substitution  we  shall  have  r-  =  — — ,~- 

or  r  -—  -^ — 7—^5  by  taking  the  sign  of  the  right  member 
of  this  equation  such,  that  r  may  be  positive. 


164 

(2.)  There  is  another  way  of  obtaining  the  preceding  ex- 
pression for  7',  which  it  may  be  proper  to  notice  in  this  place. 

Thus,  from  p.  129  we  may  clearly  represent  the  normal  at 
any  point  of  the  curve  y  =  F  (a?),  whose  co-ordinates  are  y 

and  a?,  by  Y  —  y=  —  '  \   or  its  equivalent, 

dx 

which  is  the  same  as  given  above  when  for  X  and  Y  we  put 
a?'  and  y' ;  consequently,  differentiating  this  on  the  supposi- 
tion of  the  constancy  of  x'  and  y\  we  shall,  as  before,  get 

r  =  ^^ — 7 — —  for  the  radius  of  curvature  of  the  proposed 

curve.  If  the  tangent  at  the  proposed  point  of  the  curve  is 
parallel  to  the  axis  of  a?,  or  the  axis  of  y  coincides  in  direc- 
tion with  the   normal,  we   shall   clearly   have  j)  =  0;  and 

r  =  — 7 .  If  the  curve  has  a  point,  such,  that  (without  re- 
gard to  p)  p'  =  d  -^  -i-  dx  =  0,  then,  by  taking  x  for  the 

independent  variable,  we  shall  have  -y^  =  0,   or  infinity, 

which  (since  r  =  infinity)  clearly  shows  that  the  circumfer- 
ence becomes  a  right  line  at  the  point  which  touches  and 
cuts  the  curve  at  the  point ;  and  of  course  the  point  is  gener- 
ally a  point  of  inflection,  agreeably  to  what  has  been  shown. 

To  illustrate  what  has  been  done,  take  the  following 

EXAMPLES. 

1.  To  find  the  radius  of  curvature  at  any  point  of  the 
logarithmic  curve  whose  equation  is  2/  =  a^,  or,  taking  the 
hyperbolic  logarithms,  log  ?/  =  a?  log  a. 


EXAMPLES.  165 

Taking  tlie  differentials,  we  have 

—  =  dx  loff  a     or    -f-  ^p  =zy  loo;  a, 
y  ^  dx      ^      -^     "="    ■ 

and  £  =y  :=  J  log  6j^  =  ^  log  a. 

Hence,  representing  log  a  by  — ,  we  shall  have 

llh 

^^l  =  t^^^^yl±.f.    and    y  =  i^,; 
consequently,  from  r  =  ^^ — y      ?  ^^  shall  have 

uc"  '   7/r  7ny       ' 

for  the  radius  of  curvature :  noticing,  that  the  center  of  the 
circle  must  (see  fig.  p.  142)  be  taken  on  the  concave  side  of 
the  curve. 

2.  To  find  the  radius  of  curvature  at  any  point  of  the 
common  cycloid. 


From  p.  150  we  have  ~  =  p  =  y , 


r 


and       f=y-         "^  -  -'• 

dx      -^ 


Hence  we  shall  have 

i?^  +  l  =  ?^     and     {p^-i-lf=^]/^S. 


X  -^  '  X  X 

.1 


consequently,  (^  +  1)'^  -^  ji?'  =  2  \^r  x  (2r  —  a?)  r=  the  ra- 
dius of  curvature.  Thus  it  is  manifest  that  the  radius  of 
curvature  equals  twice  the  corresponding  normal  of  the 
cycloid ;  so  that  (see  the  fig.  at  p.  149)  the  radius  of  curva- 


166  EADII  OF  CURVATURE,   ETC. 

ture  at  G  equals  twice  the  chord  BF  of  the  arc  of  the  gener- 
ating circle,  which  corresponds  to  the  cjcloidal  arc  GO. 

3.  To  find  the  radius  of  curvature  in  the  parabola,  whose 
equation  is  if  =  ^mx. 

By  taking  the  differentials  we  have  '^ydy  =  4:vidx,  which 

du  2m  ,     „  ^^  '     dp        ,  2?np 

g,ves  ^  =^=  -  ,   and    from   this  £=p  =  -  -^  . 

Hence  f +  1=^ +l  =  ^J^+^ 

y  y 

gives  (/  +  lj*  =  ^ p-^^, 

and  thence  r  =  — — -. — —  =  - — -^ —  —  ,  or  takinor  it  with 
p  —  2myj? 

the  positive  sign,  we  have  r  =  - — ^y -~  for  the  radius  of 

curvature  ;  or,  since  />  =  —  ,  we  have 

y 

_  (4:7)1^  H-  fr  _  o  ('^^  +  '^'^y  _  (normal)* 
~         4cm^        ~  m^         ~~      4w^ 

(See  Young's  "  Differential  Calculus,"  p.  131.) 

4.  To  find  the  radius  of  curvature  in  the  ellipse,  whose 
equation  is  a^f  +  h'or^  =  d/W  ;  a  and  J  representing  the  half 
major  and  minor  axes. 

Differentiating,  we  have  (£'yp  +  J^a?  =  0,   or  j9  = ^ , 

which  gives 

,  _  _  J^  ^2!^  _  _  h^^f_  _  _  l\aY--¥'b^3^)  _ _  _^ 
^  ~       a-y      a-y'^  a^y     ~  a^y^         ~      crif 

Hence,  ^-  + 1  =    ^  ,  , — ,  and  thence  (p-  + 1)^  =  -^^—t-^—^  ] 
consequently,  the  radius  of  curvature   r  =  ^   ^   iu~~     ' 


USEFUL   FORMULAS.  167 

and  since  c^y^  —  Iryp'  =  —  a-I/^  is  the  equation  of  tlie  hyper- 
bola, its  radius  of  curvature  is  evidently  of  the  same  form. 

From  what  is  shown  on  p.  128,  ^j^  =jpy  —  the  subnormal, 

and  from  a^j}^i/  -\-h'^x=  0  we  have  py  =  ——j  =  the  sub- 
normal ;  and  thence,  from  what  is  shown  at  the  same  place, 
we  have 

consequently,  from  substitution,  we  have  r  =  —p-  '  noticing, 
since  y^  =z  —^  (a^  —  a?"),  that  N  may  be  written  in  the  form 

or,  according  to  custom,  representing g—  by  e^,  we  shall 

have  ^  ^^-{a'-e'x'f. 

Substitutinsf  this  value  of  iN"  in  /•  =     ,.   ,  we  shall  have 

r  z=i — -  .  in  which  (a^  —  e^x^y  =  the  semidiameter  of 

ao  ^ 

the  ellipse,  parallel  to  the  tangent  passing  through  the  point 

of  contact  of  the  circle.     See  Young,  p.  132. 

Because  x^  =  -jj  {1/  —  y%  we  readily  get 

am  a  ' 

consequently,  if  N  makes  the  angle  L  with  the  major  axis 
of  the  ellipse,  we  shall  clearly  have  y  :=  N  sin  L,  and  thence 


168  USEFUL  FORMULAS. 

^^  |/(//+aVN'^siQ^L) 
a  ' 

wliich  gives        N  = r-  (i  —  6^  sin^  L)"*. 

Substituting  this  value  of  N  in  r,  we  shall  readily  get 


¥      __       g  (1  -  ^) 

a  {1-e'  sin^  L)'   ~  ^-e'  sin^ L)^ ' 


a  formula  that  is  very  useful  in  determining  the  figure  of  the 
earth,  L  being  the  apparent  latitude  of  the  point  of  contact 
of  the  circle  with  the  ellipse.     See  Young,  pp.  132  and  133. 

Eemarks. — 1.  The  radius  of  curvature  r  =  ,  may 
be  put  under  several  different  forms,  which  are  often  used. 
Tbis,  smce  i?  =  -r^,  we  have^-+l  =  -~j:^ —  ?  which  gives 

(f+lf  =  ^'^^'^'^'^  ,  and  from  p' =  d'^-r  ilx,  this  be- 

A     A-  'A-      -u       '    {p'  +  ^f      W^-dx^f 
comes,  after  dividmg  by  p\  — — 7—^  =  ^-^ 7—-  . 

^  dx'd  5^ 

dx 

From  what  is  shown  at  pp.  125  and  126  (see  fig.  at  p.  125), 

it  is  manifest,  since  SQ  =  dx  and  GR  =  df/  are  common  to 

the  tangent  line  and  curve,  that  SR  =  Vdy-+  dx^  must  be 

the  differential  of  the  right  line  TS   and  the   curve  AS. 

Hence,  if  s  represents  an  arc  of  a  curve  and  ds  its  diffcr- 

^    ..    ^                   W  +  cU)^          d^^        .      . 
ential,  we   shall   have    r  =  ^^ 7—^  = r-   for  its 

dx  dx 

radius  of  curvature ;  in  which,  without  destroying  its  gener 

ality,  W3  may  for  d  -~  take  its  differential  oa  the  supposi- 


USEFUL   FORMULAS.  169 

tion  that  eitlier  dx  or  dy  is  constant,  or  a?  or  y  taken  for  tlie 
independent  variable. 

Thus  since        d"^  -  ^V^^-^^^V 

ds^ 
we  shall  have  r  = 


d^ydx  —  d^xdy ' 

which,  by  taking  x  for  the  independent  variable,  reduces  to 

ds^        . 
r  =  -,   since  dx  =  const,  gives   d^x  =z  0 ;   and  if  y  is 

cL  ycLx 

ds^ 
taken  for  the  independent  variable,  it  becomes  r  =  — ~p~f~  • 

If  we  take  ds  for  the  independent  variable,  we  shall  have 
d]^  +  dx^  =  const,  and,  of  course,  its  differential  gives 

dydy"^  +  dxdrx  =  0 ;     which  gives     d^x  = 4     » 

^  dxd^x 

-^  ~  dy' 

From  the  substitution  of  these  values  in  d^ydx  —  d^xdy^ 

we  have  d^ydx  —  d^xdy  =    ^     X  d^y  =       ,  ^,    and 

d?ydx  —  cT'xdy  = -^ — ;  consequently,  from  the  substi- 

tution  of  these  values  in  r  =  -^^ — = — — ,.,    ,  ,  we  shall  have 

dyax  —  dxdy 

dxds  ,  d'lids 

r  =  -75—,     and    r  = ^r— . 

d^y  '  d^x 

2.  By  referring  to  the  figure  given  at  p.  125,  it  is  manifest 

from  the  nature  of  the  right  line,  that  if  we  pass  along  the 

tangent  and  assume  SQ  to  be  constant,  EQ  =  dy  will  also 

be  constant,  while,  if  we  pass  along  the  curve  concave  to  the 

axis  of  X  and  suppose  dx  to  be  constant,  KQ  —  dy  will 


170  E VOLUTES,  ETC. 

is  clear  that  d^y  and  -y^  must  be  positive,  when  the  con- 
vexity of  the  curve  is  turned  toward  the  axis  of  x. 

Hence,  because  the  radius  of  curvature  must  always  be 

positive,  it  is  clear  that  in  applying  r  =  ^^ — -, — '  and  the 

preceding  derived  formulas  to  practice,  ])'  and  cVy  must  be 
taken  with  the  negative  sign  in  them  when  the  curve  is 
concave  toward  the  axis  of  »,  and  with  the  positive  sign 
when  the  convexity  is  turned  toward  the  axis  of  x. 

(3.)  Eesuming  the  equation  y  =  F  (a?),  and  the  equations 
{y  —  y')'p-\-x-x'  r=^,  {y  —  y')p'=  —  (y  +  1),  from  page 
163,  it  is  manifest  that  if  we  find  x  and  y  from  any  two  of 
these  in  terms  of  x'  and  y'  and  known  terms,  that  by  sub- 
stituting them  in  the  third  equation  we  shall  have  an  equa- 
tion between  x'  and  y',  which  mil  be  the  equation  of  the 
curve  in  which  the  centers  of  all  the  radii  of  curvature  of 
the  proposed  curve  must  lie. 

Where  it  is  to  be  noticed,  that  the  equation  y  =  F  (a?)  is 
called  the  iiivolute  of  the  curve  thus  found  ;  which  is  called 
the  evolute  of  y^=¥  (x),  or  of  the  involute. 

The  reason  for  these  denominations  is  plain,  from  the  cir- 
cumstance that  we  may  regard  the  involute  as  being  generated 
by  the  unlapping  of  a  thread  placed  in  contact  with  the 
evolute,  in  such  a  way  that  the  part  unlapped  at  any  point 
equals  the  corresponding  radius  of  curvature,  when  its  ex- 
tremity will  be  in  a  point  of  the  involute.  Where  it  is 
maniiest,  that  the  radius  of  curvature  is  always  a  tangont  to 
the  evolute,  and  constantly  perpendicular  to  a  tangent  to  the 
involute  at  its  extremity. 

For  convenience  in  practice,  we  may  give  the  last  two  of . 
the  preceding  equations  the  forms 


EVOLUTES,   ETC.  171 

y'  =  y+^^    and    .'  =  .-^j+l), 
wHcli  may  be  freed  from  jp  ^-—^  and  ^'  =1  cl~  —  dx^  by 

finding  their  values  from  2/  =  F  (u?),  the  equation  of  the  in- 
volute, or  that  of  the  proposed  curve ;  when  we  may  proceed 
as  directed  above. 

To  illustrate  what  has  been  done,  take  the  following 

EXAMPLES. 

1.  To  find  the  evolute  of  the  parabola,  whose  equation  is 

y-  =z  4:mx. 

Here  for  2/  =  F  (a?),  we  have  y^  =  4:mx ;  which  gives 

di/      2  m         T       ,        ^dy       ,  2  pm  ^iv^ 

p—-f-  =  —    and    p  =d-~-~  dx= "—■  = ^ . 

■^        ax       y  -^  ax  y  y^ 

Hence  we  have  ;/  -j-  1  = „ ,  and  thence 

y 

,  1/  xy  (7ny"^Y 

and     y  =  —  {4:m-y')'\     We  also  have 

,              /if          \  2m       «         ^                        x'  —  2m 
^  =""+  \i^^  +  2/j  —  =  ^-^  +  2/^i     OY    x= —  ; 

consequently,  by  equating  these  values  of  a',  we  readily  get 

4 

y''^  =  ^  {x'—  2my-~  m  for  the  equation  of  the  evolute,  which 

is  of  the  form  of  the  well-known  equation  of  the  s^micuhiccd 

parahola     If  the  origin  of  the  co-ordinates  is  moved  in  the 

direction  of  x  positive  through  the  distance  2???.,  or  that  we 

put  x'  —  2m,=^x  and  use  y  for  y\  the  equation  of  the  evo- 

4.^-3 
lute  may  be  more  simply  expressed  by  the  form  y^  =  -^ . 

Thus,  let  CAD  represent  a  parabola  having  AB  for  its 
axis,  A  for  its  vertex,  and  E  for  its  focus ;  then,  by  setting 


172  -  E VOLUTES,   ETC. 


off  El  in  tlie  direction  of  x  positive  equal  to  AE,  it  is  clear 
that  we  shall  have  AI  =  2/7i  for  the  radius  of  curvature  of 
the  parabola  at  its  vertex,  which  equals  .4?7^  -^  2,  or  half  its 

principal  parameter,  or  latus  rectum.  Then,  since  'if-  =  ^=— 
gives  2/  ^  ±  |/  -r^— ,  we  construct  the  curve  HIK,  having  1 

for  its  vertex  by  setting  off  y  =  y   ~^~  at  the   distance  x 

/~4^ 
above  IB,  and  ?/  =  —  y  -^—  at  the  distance  x  below  IB ; 

consequently,  a  curve  passing  through  all  the  points  thus 
found,  wiil  represent  the  evolute  of  tlie  parabola,  or  the  semi- 
cubical  parabola. 

Supposing  the  evolute  to  be  correctly  constructed,  then  a 
thread  stretched  from  A  to  I,  and  lapped  on  the  branch  IK 
so  as  to  coincide  with  it,  and  made  fast  at  its  unlimitedly 
remote  extremity,  when  unlapped,  by  moving  the  extremity 
A  toward  C  and  keeping  it  stretched,  the  point  A  will  clearly 
describe  that  part  of  the  parabola  represented  by  AC.     By 


E VOLUTES,    ETC.  173 

lapping  tlie  thread  on  IH  instead  of  IK,  we  may  in  like 
manner  describe  the  branch  of  the  parabola  represented  bj 
AD. 

Remaek. — Mr.  Young,  at  page  140  of  his  "  Differential 
Calculus,"  says  that  the  evolute  does  not  extend  on  the 
side  of  X  negative,  or  from  I  toward  A,  since  x  negative  in 

y-  ■=  ^=—  will  make  y  imaginary,  which  is  undoubtedly  true ; 

yet  from  the  first  form  v'^  =  —^—p^ of  the   evolute, 

which,  for  a?  =  0  in  the  second  form,  gives  x'  =  2r/i,  clearly 
shows  that  the  point  A  is  so  connected  with  the  evolute,  that 
AI  must  be  taken  in  conjunction  with  it,  as  has  been  done 
in  the  preceding  construction ;  and  it  is  manifest  that  like 
observations  will  be  applicable  in  all  analogous  cases. 

2.  To  find  the  evolute  of  the  common  cycloid. 
From  Ex.  2,  at  p.  165,  we  have 

dy  /2r  —  x       .      ^       2r 

ax      -^        ^        X  a? 

T 

and  i>'  = -^ sr  ; 

hence,     v'  ==  V  +  - — 7~     and    x'  =^x  —  ^  ^  , , 

from  p.  171,  will,  by  substitution,  become 

y'  —  y  —  2  V2rx  —  x'     and     x'  =  x  -{-  4:r  —  2x  =  4:r  —  x. 

Hence,  from  the  substitution  of  the  value  of  y,  from  p.  150, 
in  that  of  y\  we  shall  have 

y  =  ver  sin-^a?  +  sin  ver  sin~^'r  —  2  |/(2ri»  —  a;^) 

=  ver  sin -^  a?  —  ^\2rx  —  a?^), 

or    y'=z  ver  sin -^  a?  —  sin  ver  sin-^a?;  which,  from  what  is 


174 


EVOLUTES,   ETC 


shown  at  p.  151,  is  the  equation  of  a  cycloid,  the  origin  of 
the  co-ordinates  being  at  the  extremity  of  its  base.  From 
x'  =  4?'  —  a?  we  have  a?  =  4r  —  x\  from  which  it  is  manifest 
(see  fig.  at  p.  149),  since  DB  =  2r  and  that  D  is  the  origin 
of  the  co-ordinates,  that  if  DB  is  produced  below  B  to  the 
distance  DB  or  2/',  and  then  x'  subtracted  from  4/*,  we  shall 
readily  get  —  a?  =  a?'  —  4r ;  which  clearly  shows  that  we  may 
change  the  origin  of  the  co-ordinates  from  D  to  the  point 
distant  2/*  below  B,  and  reckon  the  positive  values  of  x  from 
the  new  origin  upward  instead  of  downward,  when  the  origin 
is  at  D. 


Hence,  supposing  ABC  to  represent  the  proposed  cycloid, 
by  removing  the  origin  of  the  co-ordinates  from  the  vertex 
B  to  E,  so  that  GE  =  (xB,  we  may  reckon  x  positive  from  E 
toward  G,  and  y'  =  ver  sin~^  x  —  sin  ver  sin-^  a?,  the  equa- 
tion ol  the  semicycloid  EC,  whose  semibase  is  EL  and  vertex 
C,  will  be  that  of  the  evolute  of  the  proposed  semicycloid  BC ; 
and  in  like  manner  the  semicycloid  AE,  whose  semibase  is 
EM  and  vertex  A,  is  the  evolute  of  the  semicycloid  BA : 


EVOLUTES,    ETC.  175 

the  proposed  semicycloids  and  their  evolutes  being  clearly 
identical. 

Eemarks. — 1.  The  cycloid  DEF  being  drawn  (as  in  the 
figure)  equal  to  the  proposed  cycloid  ABC,  it  is  evident  that 
the  semicycloid  EF  and  ED  will  be  evolutes  of  EC  and  E  A ; 
and  so  on,  indefinitely,  for  semicycloids  that  may  be  de- 
scribed below  the  cycloid  DEF,  like  EC  and  EA  below  the 
cycloid  ABC.  And  it  is  easy  to  perceive  that  a  series  of 
cycloids  may  in  this  way  be  continued  indefinitely,  both 
above  and  below  the  proposed  cycloid  ABC. 

2.  To  describe  the  involutes  by  the  evolutes,  we  take  a 
thread  equal  to  the  semicycloid al  arc  EC,  and  fasten  one  of 
its  extremities  at  E ;  then,  having  lapped  it  on  the  arc  EC, 
we  carry  the  extremity  C  from  C  through  B  to  A,  when  the 
cycloidal  arc  CBA  will  evidently  have  been  described. 

To  describe  the  arcs  EC  and  EA,  we  use  two  threads  tied 
to  the  points  D  and  F,  equal  in  length  to  the  arcs  DE  and 
FE ;  then,  the  extremities  at  E,  being  carried  from  E  to  A 
and  C,  will  describe  the  semicycloidal  arcs  AE  and  EC. 

3.  To  find  the  equation  of  the  evolute  of  the  ellipse. 
From  p.  166,  we  have  c^if'  +  V^x-  =  a'lP'  for  the  equation 

of  an  ellipse,  and 

V'x  ,       ,  ¥ 

and  P+^  =  -^~^ — ,     and    ^,  = -^  . 

Hence,  from  p.  171,  the  equations 

y'  =  y  +  Pl+l    and    a,'  =  «-^(£jLl), 
become 


176  EVOLUTES,   ETC. 

From  the  equation  of  the  ellipse  we  have  ay  =  a^h'  —  5V, 
which,  substituted  for  ay  in  the  value  of  y\  reduces  it  to 

y'  —  y-  ^-^ -^^ ^-^ ,  which,  putting  a'- V^ (?^ 

is  easily  reduced  to 

,_  y  {a>  —  (?d^)  _       <?  [d?  —  ic')y_        cY  , 


y  =y- 


a'b'         ~  a'b'  b 


,4     > 


and,  in  a  similar  way,  we  have  a?'  =  — j- . 

Hence,  we  readily  get  ! 

6      ^V  o      /by'\^  ,       ,       (a'x''\^ 

y'  =  -jr    or    f=[~i-),     and    ^^  =  [-r)  ; 

consequently,  from  the  substitution  of  these  values  of  y^  and 

ar^  in  ay  +  h'x^  =  a^b\  we  have  a^  (^')    +  5^  (^-)  =  a=5= 

or  (%'0^  +  (a'^^'')*  =  ^y')^  H-  (aa^O^  =  ^*, 

and  of  course  (JyO    4-  (ax'y  =  (a^5-)*  is  the  equation  of  the 

evolute  of  the  ellipse. 

By  putting  x'  =  0,  the  equation  reduces  to 

a' 
or  its  equivalent  hy^  =  a?  —  ¥,  which  gives  y'  z=z—-  —  b]  and 

in  like  manner,  by  putting  y'  =  0,  the  same  equation  gives 

^' 

X  =  a . 

a 

Thus,  let  AB  =  2a  and  CD  =  2h  be  the  major  and  minor 

axes  of  the  ellipse,  and  let  the  points  c  and  d  be  taken  on 

the  minor  axis  at  the  distances  j-  —b  from  the  center,  while 

a  and  b  are  taken  on  the  major  axis  at  distances  equal  to 

b'^ 
a from  the  center ;  then,  curves  drawn,  as  in  the  figure, 


177 


witli  their  convexities  toward  the  axes,  so  as  to  touch  them 
at  their  extremities,  will  represent  the  evolutes  of  the  ellipse. 

It  is  manifest  that  -y  and  —  are  the  radii  of  curvature  of  the 
0  a 

ellipse  at  the  extremities  of  the  minor  and  major  axes. 

The  ellipse  may  be  described  by  means  of  its  evolute  as 
follows : 

Take  a  thread,  in  length  equal  to  the  arc  cb  +  BJ,  and 
fasten  one  end  of  it  at  (?,  and  lap  it  on  the  arc  cJ,  and  bring 
down  the  remaining  part  of  it  to  B ;  then,  carry  the  thread 
around  from  B  to  A,  and  its  extremity  B  will  describe  the 
half  of  the  ellipse  represented  by  BDA ;  and  it  is  manifest 
that  having  fastened  the  extremity  of  the  thread  at  <:7,  we 
may  in  like  manner  describe  the  remaining  half  of  the  curve, 
represented  by  BOA. 


Because  the  arc  c5  +  B6  =  the  arc  cb  + 


J- 


:'D  = 


It 


follows  that  the  arc  cb^=  i = -. —  ; 

0       a  ao 


consequently, 


since  the  four  branches  of  the  evolute  are  clearly  equal  to 

a^  —  ¥ 
each  other,  we  shall  have  4 % — •  for  the  entire  length  of 

the  evolutes.     Hence,  if  J  is  very  small  in  comparison  to  a^ 

it  is  clear  that  the  arc 
8* 


will  differ  but  little  from  ^  ;  con- 


173  EVOLUTES,    ETC. 

sequently,  the  points  c  and  d  will  fall  ultimately  without  the 
ellipse,  and  the  semi-ellipses  BDA  and  BCA  will  have  for 
their  limits  arcs  of  circles  whose  centers  are  at  c  and  c/,  and 
are  drawn  through  the  points  B,  D,  A  and  B,  C,  A. 

Remarks. — 1.  Resuming  the  equation 

of  the  evolute  from  p.  176,  then,  since  the  equation 

ay  +  Px''  =  (jt-V' 

becomes  a- if  —  V^x^  =  —  <rlr^ 

the  equation  of  an  hyperbola,  by  changing  1?  into  —  1?^ 
it  clearly  follows  that  if  for  V-  we  put  —  }p-  in  the  preceding 
equation  of  the  evolute,  it  will  be  reduced  to 

or  its  equivalent      {ax'Y  =  (cr  +  Ir^y  +  (Z>y')*' 

the  equation  of  the  evolute  of  the  hyperbola.     If  we  put 

y'  z=z  0,  the  equation  reduces  to 

ax'  =z  a^  -\-  ^',     or     x'  =  a  -] , 

a 

which  clearly  shows  that  —  equals  the  radius  of  curvature 

at  the  extremity  of  the  major  axis  (2a)  of  the  hyperbola. 

By  assuming  y\  we  can,  from  the  above  equation,  evidently 
find  the  corresponding  value  of  x' ;  and  in  this  ^ay  find  any 
number  of  points,  at  pleasure,  of  the  evolute. 

2.  It  is  easy  to  perceive  that  we  can  not,  from  the  preced- 
ing equation,  find  the  evolute  of  the  conjugate  hyperbolas ; 
which  clearly  shows  that  their  evolute  is  different  from  that 
which  has  been  found. 

To  find  the  evolute  of  the  conjugate  hyperbolas,  we  must 
proceed  in  much  the  same  way  as  before,  by  regarding  h  as 


RADII,    ETC.,    IN   POLAR   CO-ORDINATES.  179 

their  principal  semi-axis,  and  a  as  its  semiconjugate ;  conse- 
quently, we  shall,  as  before,  have  ipyy  =  (a^  +  Irf  -f  {ax' f 
for  the  equation  of  the  evolute  of  either  of  the  conjugate 
hyperbolas.     By  putting  x'  ^0  in  this  equation  we  readily 

2  2 

get  hy'  =  a-  +  §-,  which  gives  y'  z=zh  -\-  y ,  and  shows  that  -y- 

is  the  radius  of  curvature  at  the  vertex  of  either  of  the  con- 
jugate hyperbolas.  By  assuming  x'  we  can,  from  the  pre- 
ceding equation,  calculate  y\  and  thence  find,  at  will,  any 
number  of  points  in  the  evolute. 

(4.)  We  will  now  proceed  to  show  how  to  find  the  radii  of 
curvature  of  curves,  whose  equations  are  expressed  in  polar 
co-ordinates. 

Supposing,  as  at  p.  131,  that  r  cos  w  =  —  r  cos  0  =  x  and 
y  sin  0)  =  r  sin  0  =  y,  by  taking  the  differentials  of 

a?  =  —  r  cos  9     and     y  =  r  sin  0, 
on  the  supposition  that  0  is  the  independent  variable,  we 
shall  have 

dx=z  —  cos  0  dr  -\-  r  sin  ddO^  dy  —  sin  ddr  +  r  cos  Odd^ 
whose  squares  added  give  dx^  +  dy'^  =  dr^  -f  r^dO\ 

.     ,  „  ^dy        ,  /      sin  Odr  -f  r  cos  ddd\ 

And  from  d  ~  =d{ -, : ) 

dx  \—  cos  ddr  -\-  r  sm  QdQl 

we  readily  get 

d^'ydx  —  ct'xdy  =  —  r^dd'  —  2drde  +  rd^rdO ; 
consequently,  since  (see  p.  168)  the  radius  of  curvature,  r\ 
{dy'  +  dx'-f  _   {dr^  +  r'^dd^f 
,  2  7  dy  d~ydx  —  d^xdy ' 

dx 
we  shall  of  course  have  for  r'  the  expression 

(_  r'dd'  _  ^dr"  +  r(^r)  dd  ' 


180  EXAMPLES. 

noticing,  that  the  expression  can  be  put  tinder  the  more 
simple  form, 

which  must  be  taken  with  the  positive  sign. 
If  N  represents  the  polar  normal^  since 

—  W 
we  readily  get      i ' 


"-==(1)-' 


cPr' 

~dd' 

to  he  taken  with  the  positive  sign  for  the  radius  of  curva- 
ture in  curves  expressed  in  polar  co-ordinates ;  noticing, 
that  r  stands  for  the  radius  vector  in  the  polar  equation. 

EXAMPLES. 

1.  To  find  the  radius  of  curvature  of  the  spiral  of  Archim- 
edes, its  equation  being  r  =  a6. 

dr  d?7' 

Since  -j^  =  a,  we  shall  have  -7-,  =  0,  and  thence 


»= m 


=  a  (1  +  Gf^f 


+  r' 

and  ^+2(|y-^g=an2  +  n 

give  r'  =  -~^ — — -  for  the  radius  of  curvature. 
2i  -\-  tf  « 

2.  To  find  the  radius  of  curvature  of  the  spiral  whose 

equation  is  r  =  aO\ 

dr 
Since      -^  =  na&^~'^.  we  have 
do 

and  W  =  a^e^^-' {n'' +  e'f  ] 


EXAMPLES.  181 

also,  since  -j-^  =zn{n  —  1)  aO''-^  we  shall  have 

Hence  we  readily  get  r  =z  a0"-i  (n^  +  0^)^  -h  (0^  +  tv"  +  n) 
for  the  required  radius  of  curvature. 

3.  To  find  the  radius   of  curvature  of  the   logarithmic 
spiral,  its  equation  being  r  =  a^. 

Since  -J-  ■==La?  log  a    and     -^  =  a^  (log  df^  we  easily  get 

.'^  +  2(1)2-. J  =  a^-^[l  +  (log«)T; 

consequently,  we  shall  get  r'  —  <2*  |/[1  +  (log  a)-],  the  same 
as  the  normal:  noticing,  that  log  a  means  the  hyperbolic  log- 
arithm of  a. 

4.  To  find  the  radius  of  curvature  of  the  curve  whose 
equation  is  r  =  a  cos  Q, 

Here  we  have  -tH  =  —  «  sin  0,  and  — ^  =:  —  a  cos  0,  and, 
of  course,   ^  —  y  y~\   -f  r^  =:  «,   and  thence  the  radius 

of  curvature   r'  =  ^r— — :->  =  ?:• 
a'  +  a^        2 

Eemaeks. — By  referring  to  the  figure  at  p.  131,  it  is  mani- 

7^  T  7^^^  dv 

I,  fest  that  g^  =  ^,  and  g^  =  _-  (N  being  the  normal), 
;  represent  the  cosine  and  sine  of  the  angle  NSr,  the  angle 
I     made  by  the  radius  of  curvature  r',  and  the  radius  vector  r ; 


182 


EXAMBLES. 


-N' 


r*  +  2 


\ddl 


and 


N^ 


r    _ 

d^ 


dr 


\ddl 


7-= +  2 


r»  +  2 


(if-^ 


dd' 


are  t"he  projections  of  tlie  radius  of  curvature  r  on  tlie  radius 
vector  r,  and  a  line  perpendicular  to  r ;  noticing,  that  the 
first  of  these  projections  is  sometimes  called  the  co-radius 
of  curvature.     Kepresenting 

dr         , 


^    \de)       difi 


\ldry        .dr\        I    ,      ^UlrV-  d}A 

by  y'  and  x\  we  shall  have 


fdr\^       ^ 


r  + 


and 


"^  +  ^  V/0/     "^  d(^ 

W0/  ' 


=  y 


dr 


/^  +  2/~ 


dh 


\dd)       ^  dO^ 


for  the  rectangular  co-ordinates  of  a  point  in  the  evolute  of 
the  proposed  curve,  whose  origin  is  the  same  as  that  of  the 
proposed  curve. 

Thus,  in  the  case  of  Example  3,  we  have  found  in  the 


EXAMPLES.  183 

logarithmic  spiral  r'  =  a^  |/[1  +  (log  a)"']  =  N,  the  normal, 

and,  of  course,  we  shall  have  r  —  '/  X  ^=  r  —  r  =  0  =y' ; 

so  that  the  center  of  the  evolute  coincides  with  that  of  the 
proposed  spiral.     And 

,        dr        ^^        dr        dr         ^  .  , 

or,  representing  x'  by  r'\  we  shall  have  r"  =  a^  log  a  ;  con- 

r" 
sequently,  since  r  —  a^  ^  we  shall  have  —  =  log  a,  a  con- 
stant ratio.  .  Hence,  since  r"  is  perpendicular  to  r,  and  has 
a  constant  ratio  to  it,  it  is  manifest  that  the  evolute  must  be 
a  logarithmic  spiral  similar  to  the  proposed  spiral,  their  radii 
vectors  making  equal  angles  with  their  arcs. 

(5.)  There  is  another  method  of  finding  the  radius  of 
curvature,  that  is  often  very  useful  in  polar  co-ordinates, 
that  may  be  noticed  in  this  place. 


t 


1.  Thus,  let  the  curve  ABC  be  supposed  to  be  described  by 
the  extremity  of  PB  =  r  during  its  angular  motion  around  P 
in  the  same  plane,  in  the  order  of  the  letters  A,  B,  C  ;  then,  if 
BKL  is  the  circle  of  curvature  at  the  point  B,  having  O  for 
its  center,  and  its  radius  OB  =  R  drawn  to  its  point  of  con- 


184:  ANOTHER  METHOD, 

tact  with  the  curve  ABC,  or  the  tangent  T^  of  the  curve  at 
the  same  point  B,  by  drawing  PD  and  PF  perpendicular  to 
BO  and  the  tangent,  we  shall  have  the  right  triangle  POD, 
which  gives 

PO-  =  PD=  +  OD-2  =  PD=  +  (BO  -  BDf 

=P]»  BO^-  2B0  X  BD  +  BD-=  PB^-  2B0  x  BD  +  B0», 

which,  since  PB  =  ;•  and  OB  =  E,  by  representing  BD.=  PF 
by  ^>,  becomes  PO^  =7^—2dB.-\-  K" ;  or,  denoting PO  by  r', 
we  shall  have  r''  =  '^  —  2v'R  +  RK 

Since  the  points  P  and  0  are  fixed  for  the  same  circle  (by 
regarding  the  curve  and  circle  as  having  a  very  small  com- 
mon arc),  we  may  take  the  differential  of  this  equation,  on 
the  supposition  of  the  constancy  of  7*'  and  R,  and  shall 

thence  get  rdr  —  Rdu  =  0  ;  which  gives  R  =  -j—  for  the  re- 
quired expression  for  the  radius  of  curvature.  Admitting 
the  construction  of  the  figure,  the  equiangular  triangles  PBD 
and  LBK  clearly  give  the  proportion  PB  :  BL  : :  BD  :  BK, 

or  its  equivalent  r  :  2R  ::v:  BK  = =  — : —  =  the  chord 

^  r  do 

of  curvature  which  passes  through  the  pole,  or  origin  of  the 

co-ordinates ;  which  is  a  result  that  is  very  useful  (as  is  the 

radius  of  curvature)  in  the  doctrine  of  central  forces.     (See 

Vince's  "Fluxions,"  pp.  149  and  242,  together  with  ISTew- 

i  ton's  "  Principia,"  vol.  i.,  p.  68,  &c.) 

There  are  one  or  two  forms  of  v  that  are  often  useful, 
which  it  may  be  well  to  notice. 

2.  Thus,  if  the  angle  PBF  made  by  r  and  the  tangent  T^,  is 
represented  by  <^ ;  the  right  triangle  PBF  gives 

PF  =  ^  =  7"  sin  <^. 

Also,  if  PGr  is  assumed  for  the  angular  axis,  and  the 


WITH  EXAMPLES.  185 

angle  GPB  =  d ;  then  we  shall,  from  tlie  principles  given  at 
p.  134,  get  v=.rx  ^^^^^^  =  -77-— /^r 

From  this  expression,  we  readily  get 
dv  =^  — 


rdr 


I 


consequently,  E  =  '-^  will  be  reduced  to  the  form 

which  agi'ees  with  the  form  of  r\  the  radius  of  curvature, 
found  at  p.  179 ;  noticing,  that  the  radius  of  curvature  must 
be  taken  with  the  positive  sign. 

Hence,  it  follows  that  the  equation  of  the  evolute  of  the 
proposed  curve  may  be  found  from  the  expressions  for  y' 
and  x\  given  in  the  remarks  at  p.  182.  It  may  be  added, 
that  having  found  E,  we  can  easily  find  PO  or  v\  from  the 
triangle  FOB,  and  also  the  angle  BPO ;  consequently,  the 
evolute  can  be  constructed  by  points. 

EXAMPLES. 

1.  To  find  the  radius  of  curvature  in  the  ellipse  when  re- 
ferred to  polar  co-ordinates,  the  origin  being  at  the  focus. 

Taking  a  and  h  for  the  half  major  and  minor  axes,  we 

have,  from  a  well-known  property  of  the  curve,  the  equation 

la  —  r      h^       ,         ,.„         ..    .        adr      h'dv       ,    , 
=  -^ ;  whose  differential  gives  —^  —  — T'  ^^^  thence 


186  EXAMPLES. 

rdr      _,       7'^      ¥ 
-T-  =  R  =  -3  X  - 


rdr  7'^      y^  h^ 

=  R  =  -3  X  - .     If  we  put  J?  =  -  =  the  semiparameter 


of  the  major  axis,  this  becomes  R  =  -3  x  ^ ;  which  is  easily 

shown  to  agree  with  the  value  of  the  radius  of  curvature 
found  at  p.  167. 

This  expression  will  enable  us  to  find  the  radius  of  curva- 
ture either  of  the  ellipse,  hyperbola,  or  parabola,  by  putting 
p  for  the  parameter  of  the  major  axis,  and  observing  that 
r  equals  the  distance  of  tJie point  of  the  curve  whose  radius 
of  cwi^ature  is  to  be  found  from,  the  focus  or  origin  of  the 
co-ordinates,  and  that  v  equals  the  perpendicular  from  the 
focus  to  the  tangent  to  the  curve  at  the  same  point. 

2.  To  find  the  radius  of  curvature  of  the  parabolic  spiral 
whose  equation  is  ?•  =  a0  . 

By  taking  the  differentials,  we  have  -j-  =  -  ad   ^,  and 

consequently,  we  shall  get 

r"  2ad^ 


A(tH 


(402  ^  l)i' 


or  shall  have  v  = .     Hence  we  readily  get 

a  (40^  +  3)  e^dd 
dv=  -^ ^-3 — ; 

.      ,  ,        a'dO  ^   ^      a  {46^  +  if 

consequently,  from  rdr  =  -^r—  we  get  K  == ,  as 

^         -^  2  ^  (4^  +  3)0^ 

required 


EXAMPLES   IN  INTERSECTING   LINES.  187 

3.  To  find  the  radius  of  curvature  of  the  equiangular  or 
logarithmic  spiral. 

From  r  sm  (f)  =  v,  since  </>  is  constant  or  invariable,  we 

Tclv 

get  dv  =  sin  (jxlr ;    and  thence  from   -y— ,  we   easily  get 

R  =  -: — -,  as  required.     It  is  hence  manifest  that  R  and  r 
sm  (f> 

are  the  hypotenuse  and  leg  of  a  right  triangle,  having  the 
angle  opposite  to  r  equal  to  </> ;  consequently  it  follows,  as  at 
p.  182,  that  the  evolute  must  be  an  equiangular  spiral  simi- 
lar to  the  proposed  spiral,  and  having  the  same  center. 

(6).  Sometimes,  as  in  finding  the  radius  of  curvature  in 
(2),  at  p.  164,  by  regarding  the  evolute  as  being  formed  by 
the  intersection  of  successive  normals  to  the  involute,  we 
obtain  a  convenient  method  of  finding  the  locus  of  the  in- 
tersections of  lines  or  surfaces  drawn  according  to  some  law, 
which  are  sometimes  called  consecutive  lines  and  curves. 

EXAMPLES. 

1.  Suppose  we  have  the  equation  x^  —  yz  ■\-a^=  0,  such 
that  z  is  arbitrary;  then  it  is  required  to  find  the  curve 
resulting  from  the  elimination  of  z  from  the  equation,  on 
the  supposition  of  the  constancy  of  y  and  x  when  z  varies. 

By  putting  the  differential  coefficient  with  reference  to  z 

equal  to  naught,  we  have  ^xz  —  y  =  0,  which  gives  s  =  -^—  . 

Hence,  putting  this  value  of  z  in  the  proposed  equation,  we 

y^        2?/^ 
have  J -^  +  a  =  0,     or    y*  =  4:aXj  the  equation  of  a 

parabola  whose  parameter  equals  4a. 

Remark. — ^If  the  proposed  equation  had  been 
xz'^  —  yz^  -i-  a  =  Oj 


188  EXAMPLES  IN  INTERSECTING  LINES. 

2?/ 
by  a  similar  process  we  should  have  found  2  =  -^ ,  and 

27 
thence  have  obtained  y^  ^  -^  aar^  the  equation  of  the  semi- 

cuhical  pa7*ahola^  for  the  result  of  the  elimination  of  2  from 
the  proposed  equation. 

2.  Given  x'-  +  y"  =  r'^  and  {x  —  xj  +  (//  —  yj  =  r» 
for  the  equations  of  two  circles,  to  eliminate  x'  and  y'  from 
them. 

By  taking  the  differentials  of  the  equations  by  regarding 
x'  and  y'  alone  as  variable,  we  have  x'dx'  +  y'dy'  =  0,  or 

dy'  x' 

-^,  = -, ,  and  from  the  other  equation  we  in  like  man- 

\Anll  y 

ner  get  ^,  = 7-7;  consequently,  from  equating  these 

J  if 

T  X  —  X 

values  we  get   —  =  7 ,    or    x'y  =  y'x^    which  gives 

a?'  =  — .     From  the   substitution  of  this  value  of  x'   in 

y 

t'v 
jjj/2  _j_  y'i  _  ^n^   ^Q  gg^    y'  _  __^_^        ^    which  reduces 

1/  X                         7*  X 
x'  =  ^—  U)  x'  =  — -j^, ^ .     Ilence,  from  the  substitution 

y  Vip'  +  y') 

of  these  values  of  y'  and  x'  in  {x  —  x'y  +  {y  —  y'f  =  r^, 
we  readily  get  x^  -i-  y^  —  2/  \/  {x^  +  //")  +  r'^  =  7'^,  or  by  ex- 
tracting the  square  root  of  both  members  of  the  'equation, 

we  have  ^(a^ -{-  y^)—  r'  =  ±  r, 

or  its  equivalent,  x"  +  y^  =  {/  ±  ry, 

equivalent  to  two  circles,  represented  by  ay^  +  y'^=  (r'  -j-  ry 
and  a^  -j-  y^  =  {r'  —  7')l  Hence,  the  series  of  circles  repre- 
sented by  (a?  —  x'y  +  (y  —  y'y  =  r^,  are  touched  on  their 
outside  and  inside,  or  said  to  be  enveloped  by  the  circles 


EXAMPLES  IN  INTERSECTING   LINES.  189 

^  -\-  y^  z=z  {^r'  +  ry-  and  a?^  +  2/'  =  i^r'  —  rf^  which  clearly 
have  the  same  center  as  the  given  circle  x'^'  +  y'~  —  r'l 

Eemark. — The  preceding  solution  is  merely  a  modifica- 
tion of  that  given  by  Young,  at  pages  146  and  147  of  his 
"  Differential  Calculus." 

3.  Supposing  ACB  to  be  a  triangle,  such  that  the  position 
of  AB  being  changed,  the  area  of  the  triangle  shall  be  inva- 
riable, then  it  is  required  to  find  the  curve  to  wliich  AB 
shall  always  be  a  tangent. 


Eepresenting  AC  and  CB  by  x  and  ?/,  and  assuming 
2/ 1=  aa?  -f  2>  for  the  equation  of  AB  when  referred  to  x  and 
y  as  axes  of  co-ordinates,  by  putting  a?  =  0,  we  shall  have 
y  =z  h    or    CB  =  7> ;    also,   by   putting   j/  =  0,    we   have 

ax  +  5  =  0,  which  ejives  x  =1 or    AC  = . 

Because,  from  the  principles  of  mensuration,  the  area  of 

,,        .       ,     .  ^-p.       AC  X  CBsinano^C  l/'miG    .^ 

the  triangle  ACB  == ■'^—  ■=  —  —^ —  ;  if  we 

represent   this   by   ^,   we  shall    have   5  =  —  — ^ or 

a=—  —^ .     Hence,   from   tlie   substitution   of  a,   the 

equation  y  =  ax  -{-  h,  becomes  y  =: -; —  x  -\-  h;  whose 

ZiS 

differential  coefficient  taken  by  regarding  h  alone  as  variable, 

gives -1-1=0,   which  gives   h  =  — .— ^ .     Sub- 

s  X  sm  \j 


190  EXAMPLES  IN  INTERSECTING  LINES. 

stituting  this  value  of  J  in  y  = x  +  h,  it  becomes 

lent  xy  =  —^—^^  the  equation  of  an  hyperbola  between  its 

asymptotes,  as  required,  since  the  curve  must  clearly  always 
touch  the  side  AB  in  all  its  positions. 


SECTION  VII. 

MULTIPLE    POINTS,   CUSPS   OR  POINTS  OF   BEGRESSION,   ETC. 

(1.)  Multiple  Points. — If  two  or  more  brandies  of  a 
curve  cross  eacli  other  at  a  point,  the  curve  is  said  to  have 
a  multiple  point  of  the  first  kind ;  the  point  being  called 
double,  triple,  &c.,  when  two,  three,  &c.,  branches  cross  at 
the  point :  also,  when  any  number  of  branches  of  a  curve 
touch  each  other  at  a  point,  it  is  said  to  be  a  multiple  point 
of  the  second  kind. 

If/*  (cc,  2/)  =  0  represents  the  equation  of  a  curve,  it  is 
manifest  that  we  may  find  its  multiple  points  of  the  first 
kind,  by  determining  those  points  of  the  curve  where  we 

have  y-rr)  =  J9":  such,  that  n  is  a  positive  integer  equal  to 

the  number  of  branches  that  cross  each  other  at  the  point, 

fly 

and  -~-  =z  p  represents  the  tangent  to  any   one  of  the 

branches  at  the  same  point 

It  is  hence  manifest,  that  to  find  p^  we  may  differentiate 
y  (a?,  2/)  ==  0  n  times  successively,  by  regarding  x  and  y  as 
being  independent  variables,  or  by  considering  dx  and  dy 
each  as  being  c(5nstant  or  invariable,  when  the  successive 
differentials  are  taken. 

EXAMPLES. 

1.  To  find  the  multiple  point  of  the  curve  whose  equation 
is  ay"^  -f  cxy  —  hx^  =  0,  at  the  point  whose  co-ordinates  are 


192  MULTIPLE   POINTS,   ETC., 

a?  =  0    and    y  =  0,   or  at  the  origin  of  tlie  co-ordinates. 
By  taking  the  successive  differentials,  we  have 

2ayd?/  +  cxdy  +  cydx  —  Shx^dx  =  0, 
and  2a<l(/^  +  Icdxdy  —  ^Ixddc^  —  0  ; 

which  gives 

a  a 


;~,  +  —f x  =  0,     or    y +  -^ ^  =  0. 

dxr       adx        a  -^        ^  ., 


By  putting  x  =  ^   in  this,  we  have  p^  +  ~  —  0,   which 

c                               c 
gives  i?  =  0,     and    j!>  +  -  =  0,  or  />  = ;  consequently, 

the  curve  has  a  double  point  at  the  origin  of  the  co-ordinates, 
one  of  whose  branches  touches  the  axis  of  a?,  since  one  value 
of  p  equals  naught,  and  the  other  branch  makes  an  angle 

G 

with  the  axis  of  x^  whose  tangent  = . 

Another  Solution. — Solving  the  equation  by  quadratics, 


V           ,                          ex        ex  (^    ,     2abx        .     \ 
'     wehave  y=--±^-(l-f  ^^ ,&c.); 


whose  roots 


hx^         .               ,                   ex      hx^ 
are  y  = ,  &c.,     and    v  = h,  &c. 

By  taking  the  differentials  of  these  values  of  ?/,  regarded 
as  a  function  of  a?,  we  get 

do       21)x         .  T     du  G       2hx         - 

— ■  = ,  &c.,     and     -y-  = h,  &c. ; 

dx         G  dx  a         G 

consequently,    putting    a;  =  0,    we  get  -^  =  ^  =  0,    and 

~  = ,  the  same  as  before. 

dx  a 

2.  To  find  the  multiple  points  of  y'^  =  (a?  —  a)V. 
By  taking  the  differentials,  we  have 


WITH   EXAMPLES.  193 

^ydy  =  2  {x  —  a)  xdx  -f-  {x  —  a)  ^cLc, 

whicli  is  satisfied  so  as  to  leave  c?y  and  dx  undetermined,  by 
putting  y  =  0  and  x  —  a  =  0  or  x  =  a;  consequently,  if 
there  is  a  multiple  point,  it  must  evidently  be  at  the  point 
represented  by  y  =  0   and  x=:  a. 

To  find  wbetber  y  =  0  and  x  =  a  correspond  to  a  mul- 
tiple point,  we  diflferentiate 

2ydy  =  2  {x  —  a)  xdx  -\-  (x  —  dfdx^ 
by  regarding  dx  and  dy  as  constant,  and  get 
2dy''  =  2xdx'  +  4  (a?  —  a)  di(^, 

which,  by  putting  x  —  a^  reduces  io  l-j^\  z=  a^  or  dy  =  \/a 

and  -~-  •=  —  \fa\  consequently,  a  double  point  exists  at  the 

point  whose  co-ordinates  are  y  =  0   and  a?  =  «. 

8.  To  find  the  multiple  points  of  the  curve  whose  equa- 
tion is  2/^  =  h^x  +  2W  -\-  arl 

Here  we  have  2ydy  =  Irdx  +  4:hxdx  +  Sx'^dx,  which  is 
satisfied  so  as  to  leave  dy  and  dx  undetermined  by  putting 
y  =  0  and  J'  -f-  4:bx  +  Sx^  =  0,  or  x=:  —  h.  Hence,  as  in 
the  preceding  example,  we  have  2dy^  =  4:hdxF  +  6xdx%    or 

(-— )  =  45  + Ga?;  or,  putting  x  =  —5,  we  have  ( -^J  =  —2h; 

^y  ^  A/— in.  .^A  ^y 


consequently,  -^  —  V  —2h  and   — ^  =  —  V—  25,  which  is 
dx  dx 

a  double  point  when  h  is  negative.  If,  however,  5  is  posi- 
tive, the  point  represented  by  y  =  0  and  x  =  'b^  must  clearly 
be  detached  from  all  the  other  points  of  the  curve,  though 
connected  with  them  by  the  same  equation;  and  such  a 
point  is  called  an  isolated  or  conjugate  point.  (See  "  Cal. 
Dif.,"  p.  101,  of  J.  L.  Boucharlat;  and  Young,  p.  150.) 


194  MULTIPLE  POINTS,   ETC., 

4.  To  find  tlie  multiple  points  of  the  curve,  whose  equa- 
tion is  y'  =  (a;  —  a)V. 

By  taking  the  differentials,  we  have 

3y  Wy  =  S(x  —  afardx  +  2{x  —  dfxdx, 

which,  bj  putting  2/  =  0  and  a;  =  a,  leaves  dy  and  dx  unde- 
termined ;  consequently,  if  there  is  a  multiple  point,  it  must 
clearly  correspond  to  y  =  0  and  x  =  a.  Hence,  taking  the 
successive  differentials,  regarding  dx  and  dy  as  constant,  we 

readily  get  [— )  =  x^^  or,  putting  a  for  a?,  we  have 

Since  this  has  but  one  real  root,  it  clearly  results  that  the 
point  corresponding  to  y  =  0  and  a;  =  a  is  not  a  multiple 
point. 

Kemarks. — ^It  may  be  shown,  in  much  the  same  way,  that 
the  equation  y"  z=z  {x  —  a^x"^,  when  n  is  an  odd  integer,  can 
not  have  a  multiple  point;  and  that  when  n  is  an  even 
integer,  it  has  a  double  point 

5.  To  find  the  multiple  point  of  y^=  (x  —  dfx. 

It  is  easy  to  perceive,  on  account  of  the  inequality  of  the 
exponents  of  y  and  a?  —  a,  that  the  curve  represented  by  the 
proposed  equation,  can  not  have  a  multiple  point  of  the  first 
kind;  consequently,  we  will  proceed  to  determine  whether 
it  has  a  multiple  point  of  the  second  kind. 

Since  by  putting  x  =  a,  the  equation  is  satisfied,  and  gives 
y  =  0,  by  taking  its  differentials  we  have 

2ydy  =  4  (a?  —  dfxdx  -\-  {x  —  dfdx^ 
which  is  also  satisfied  by  putting  a?  ==  a  and  y  —  0,  and  by 
taking  the  differentials  of  this  by  supposing  y  to  be  a  func- 
tion of  x  or  dx  constant,  we  have 


WITH   EXAMPLES.  195 

2yc^V  4-  2iy=  =  12  (;»  -  afxdj(r  +  8  (aJ  -  afdxP, 
wliicli  is  satisfied  by  putting  x  =  a,  y  =  0,  and  dy  =  0,  which 
leave  drt/  undetermined.     By  taking  the  differentials  as  be- 
fore, we  have 

2(/d^y  +  Myd^y  =  24  (a?  —  a)  xdx^  +  36  (a?  —  afdx^^ 
which  is  also  satisfied  by  putting  x  =  a,  y  =z  0,  dy  =^0,  and 
leaves  d^y  undetermined. 

Taking  the  differentials  of  this,  we  have 
2ijd'y  +  Sdyd'y  +  6  (dyf  =  24:xdx'  +  96  (a?  -  a)  dx^ ; 
which,  by  putting  x  =  a,  y  =  0,  dy  =  0,  is  reduced  to 
6  (d'^yf  =z  2^adx\ 

and  is  equivalent  to  (y'^j  =4a,  or,  extracting  the   square 
roots  of  the  members  of  this,  we  have 

3  =  V«     and     g=-2ya. 

It  is  hence  evident  that  the  curve  has  two  branches  that 
touch  the  axis  of  x  on  opposite  sides,  and  each  other  at  the 
point  whose  co-ordinates  are  x  =  a  and  y  =  0,  since  dy^=0 

or  -^  =  0 ;  and  that  the  order  of  contact  of  the  branches 
dx 

with  the  axis  of  a?,  and  with  each  other,  may  be  expressed 

Otherwise.  By  taking  the  square  root  of  the  members 
of  the  proposed  equation,  we  shall  have  y=:{x  —  ay  \/x. 
Which,  by  taking  the  differentials  of  its  members,  gives 

-£  =  2{x-a)^x  +  ^{x-  afx~\ 
which,  by  putting  x  =  a^    gives    -^  =  0. 


196  MULTIPLE  POINTS,   ETC., 

Hence,  taking  the  differentials  again,  gives 

which,  by  putting  x  =  a,   reduces   to  -^  =  ±:2\/a',  since 

the  square  root  ought  to  be  taken  with  the  ambiguous 
sign  ±.  Hence,  as  before,  two  branches  of  the  curve 
touch  the  axis  of  x  on  opposite  sides,  and  each  other  at  the 

point  (y  =  0,  x  =  a)  with  contact  of  the  order  -~  ,  and  of 

course  the  curve  has  a  double  point  of  the  second  kind  at 
the  point  (y  =  0,  x  =  a). 

Remarks. — It  is  manifest  that  this  process  is  more  simple 
than  the  preceding.  And  it  is  manifest  that  in  either  method 
we  may  take  the  differentials  of  the  right  members  of  the 
equations  (since  it  will  not  affect  the  results),  without  taking 
that  of  37,  the  factor  of  {x—a)\  (x—af,  &c. 

6.  To  find  the  multiple  point  corresponding  to  3/  =  0  and 
a;  =  a  in  the  curve  whose  equation  is  y  =  {x  —  af  \/x. 

Since  this  curve  evidently  can  not  have  a  multiple  point 
of  the  first  kind,  we  proceed  to  determine  the  multiple  point 
of  the  second  kind  (by  regarding  \^x  as  constant),  as  in  the 
otherwise    of   the   preceding   example.      Hence,   we    have 

dy  =  S  {x  —  af\/xdx^  which  for  x  ^=  a  gives  -.--  =  0,  and 

shows  that  the  curve  touches  the  axis  of  x  at  the  point 
(y  =  0,    a?  =  a).     Taking  the   differentials   again,   we  have 

■3^  =  6  (a?  —  a)\/x^  which  x  =  a  reduces  to  -y^  =  0,    and, 

of  course,  the  curve  has  contact  of  the  order  — ^  =  0  at  the 

dx^ 

point  (j/  =  0  and  x  =  a).     By  taking  the  differentials  again, 


WITH  EXAMPLES.  197 

we  liave  -7^  =  6|/a?,  which,  by  putting  a  for  a?,  and  taking 

±  before  the  square  root,  gives  -~-^  =  ±  6\/a ;  consequently, 

the  proposed  curve  has  a  double  point  of  the  second  kind,  at 
the  point  (y  —  0,  x  =  a),  whose  order  of  contact  is  expressed 

Remarks.— 1.  If  y  =  (x  —  a)" a?"*-}-  h,  in  which  m  and  n 
are  positive  integers,  then,  it  is  clear  that  we  may  pro- 
ceed in  the  same  manner  as  heretofore  to  find  the  multiple 
*  points. 

Thus,   if  71  =1,   by  taking  the   differentials   we   havo 

df/         1         /dyY  ,         ■  n  -.  P 

-^  =  a;"*,  or  l-~j  =  a?;  and  puttmg  a  for  a?,  as  heretofore, 

(dy\"^ 
~j  =0,  which,   when  m  is  an  even  number, 

gives  (-—I  —  ±  t^a;   which  clearly  gives  a  double  point 
of  the  first  kind,  at  the  point  (a?  =  a,  and  y  =  !>)',  noticing, 


if  m  is  an  odd  integer,  that  ^  =  |/a  is  not  a  multiple,  but 

a  single  point,  since  "^/a  can  not  have  but  one  real  odd  root, 

the  remaining  roots  being  repetitions,  imaginary  or  impossible. 

If  n  is  greater  than  1,  and  771  odd,  the  curve  will  have  a 

single  point,  the  order  of  contact  being  expressed  by  -^r'i  i 

but  if  m  is  even,  the  curve  will  have  a  double  point  of  the 
second  kind,  expressed  by  ±  |/rt,  at  the  point  (x  =  a  and 
y  =zhy^  see  Young,  pp.  151, 152  ;  observing  that  Mr.  Young 
is  clearly  incorrect,  when  he  says  that  a  radical  of  the  third 
degree  gives  a  triple  point,  and  a  radical  of  the  7;7tli  dcgreo 


198  CUSPS,   OR  POINTS  OF  REGRESSION. 

will  indicate  that  m  branches  of  the  curve  meet  at  the  point 
(a?  =  a  and  y  =  J). 

2.  If  y  (a?,  y)  =  0  represents  an  explicit  function  of  x 
and  y,  then,  by  finding  y  in  terms  of  a?,  after  the  manner 
of  solving  equations,  and  then  proceeding  as  before,  we  may 
find  the  multiple  points,  as  above. 

(2.)  Cusps^  or  Points  of  Regression. — -A  cusp,  or  point 
of  regression,  is  generally  considered  as  a  species  of  double 
point,  at  which  two  touching  branches  of  a  curve  stop  or 
terminate.  If  the  convexities  of  the  branches  touch  each 
other,  the  point  is  called  a  cusp  of  the  first  kind;  while,  if 
the  concavity  of  one  branch  is  touched  by  the  convexity 
of  the  other,  the  point  is  said  to  be  a  cusj}  of  the  second 
kind. 

It  is  evident,  that  the  particular  co-ordinates  of  points 
where  cusps  exist  must  be  found  by  particular  considera- 
tions, and  not  by  the  application  of  Taylor's  Theorem ;  for 
otherwise  the  branches  would  be  continued  through  the 
cusp,  and  make  it  a  multiple  point,  instead  of  a  cusp; 
against  the  hypothesis. 

Remark. — When  more  than  two  branches  of  a  curve 
touch  each  other  and  stop,  it  is  plain  that  we  may  regard 
them  as  being  cusps,  and  proceed  to  treat  them  in  the 
same  way. 

EXAMPLES. 

1.    "To  find  the  cusps  of  the  curve  whose  equation  is 
y^  =  x'{l-  xj  =  x^  -  Sx'  -f  3aj«  —  x^V' 

It  is  manifest  that  the  equation  is  satisfied  either  by  put- 
ting x  =  0  and  y  =  0,  or  by  putting  x=  ±1  and  y  =  0. 
Hence,  to  determine  which  of  these  gives  cusps,  we  may 
take  the  differentials  of  the  members  of  the  equation  on  the 


CUSPS,    OR   POINTS   OF   REGRESSION.  199 

supposition  of  tlie  constancy  of  dy  and  dx  in  the  successive 
dilierentiations.     Hence  we  shall  have 

ydy  =  (2.»^  -  9a^  +  12^^  -  5x')  dx, 

which  is  satisfied  hj  x  =  0,  or  cc  =  it  1  and  y  =  0,  while 
dy  and  dx  remain  undetermined.  By  taking  the  differen- 
tials again,  we  have 

dy""  =  {6x^  -  4:5x^  +  84:x'  -  'iBx^)  dx\ 

which,  by  putting  iz;  =  0    or    a?  =  ±  1,  gives 

consequently,  by  extracting  the  square  root,  we  have 

^  =  0,     and     ^=-0, 
dx  ax 

both  when  a?  =  0   and  when  a?  =  ±  1. 

It  is  hence  clear  that  the  axis  of  x  is  touched  on  opposite 
sides  at  the  origin  of  the  co-ordinates,  and  at  the  extremities 
of  the  axis  of  a?,  represented  by  a?  =  ±  1?  or  by  a?  =  1  and 
«  =:  —  1,  on  the  positive  and  negative  sides  of  the  axis. 
Where  it  is  manifest  that  the  extremities  of  the  axis  must 
be  cusps  of  the  first  kind,  since  the  convex  branches  of  the 
curves  touch  the  axis  of  x  and  each  other  at  the  extremities 
of  the  axis,  and  stop  at  those  points.  It  is  also  plain  that 
a?  —  0  and  y  ^=^^  correspond  to  a  double  point  of  the  second 
kind,  since  the  curve  is  evidently  continued  through  the 
origin  of  the  co-ordinates. 

Otherwise. — Eesuming  the  equation  ?/^  z=  x^  (1  —  a?-y,  to 
find  its  cusps  we  may  differentiate  successively  y^  and 
(1  —  a?Y  without  a?*,  or  by  regarding  x  as  constant,  except  so 
far  as  it  is  contained  in  I  —  a?l     Hence  we  have 

^ydy  rrr    -   6^  (1  -  xjdx  ] 


200  CUSPS,   OR  POINTS  OF  REGRESSION. 

and  differentiating  again  with  reference  to  (1  —  a?^/,  as  before, 

we  have         2  {y(Py+df)  =  2^^  (1  -  ar*)  c^ ; 

and  differentiating  again  with  reference  to  1  —  ar*,  we  have 

2  {ycPy  +  ^dyd'y)  =  -  ^Sx^da^. 
By  putting  x  =  ±1  and  y  =  0  in  this  equation,  we  have 
dy  ^y  _       ^  . 

consequently,  the  curvatures  at  the  points  a'  =  ±  1  are  of 
higher  orders  than  as  found  by  the  preceding  method,  and 
they  are  cusps  of  the  first  kind. 

To  find  the  multiple  point  corresponding  to  a?*,  we  may 
reduce  j^  =  a;*  (1  —  aj^)  to  y  =  x^  (JL—  x^)%  and  take  the 
differentials  with  reference  to  ar'  by  regarding  (1  —  ar*)^  as 
being  constant     Ilence  we  shall  have 

dy  =  2x{l—  Qpf  dx  ; 
and  in  like  manner,  d-y  =  2  (1  —  x^f  ds?' ; 

^X  =2'{i-xJ. 

Putting  a?  =  0  in  this,  we  have 

consequently,  the  curve  has  a  multiple  point  of  the  second 
kind,  at  the  origin  of  the  co-ordinates  ;  the  curvature  being 
clearly  of  the  sec'ond  degree. 

2.  To  find  the  cusps  of  the  curve  represented  by 
y  =:  x^  -^  x^. 

The  equation  is  satisfied  by  putting  x  =  0  and  y  =  0, 
or  at  the  origin  of  the  co-ordinates ;   and  by  taking  the 

differentials,  we  have  -j-=  2^  +  ^  a?^,   which,  by  putting 


201 

du 
cc  =  0,  gives  -^  =:  0.     It  is  hence  clear  that  the  curve  has 
°         dx 

a  cusp  of  the  second  kind  at  the  origin  of  the  co-ordinates, 

its  two  branches  touching  the  axis  of  x  and  each  other  at 

the  origin. 

3.  To  find  the  cusps  of  the  curves  represented  by 
y-  =r  ±  a^. 

The  equations  are  satisfied  by  aj  =  0  and  2/  =  0  at  the 
origin ;  and  by  taking  the  differentials,  we  have 

which,  by  taking  the  differentials  again,  regarding  dy  and  dx 
as  constant,  gives      2  (^J  =  ±  6,»; 

which,  by  putting  a?  =  0,  gives  -y-  —  ±  0. 

u,x 

It  is  hence  clear  that  the  carves  have  cusps  of  the  first 
kind  at  the  origin,  the  convexities  of  the  curves  touching 
each  other. 

The  cusps  may  be  represented  by  0<  and  >0,  in  which 
0  is  the  origin,  the  positive  values  of  x  being  reckoned 
toward  the  right ;  and  the  first  figure  corresponds  to  the 
sign  +,  while  the  second  corresponds  to  the  sign  — ,  in  the 
proposed  equations. 

4  "To  find  the  cusps  of  the  curve  expressed  by 
{y-lf  =  {x-af." 

The  equation  is  clearly  satisfied  by  y  =  5  and  x  =  <z,  and 
by  taking  the  differentials 

dy  _2    X  —  a         2        1 

and  putting  x=^a^  we  have  -—  =  infinity,  or  —  =  0 ;  conse- 
9* 


202  CUSPS,  OR  POINTS  OF  REGRESSION. 

quently,  setting  off  a?  =  a  from  the  origin,  and  drawing  tlie 
perpendicular  y  =  h  to  the  axis  of  x  through  the  point  thus 
found,  by  putting  x  =  a  +  h  vfe  shall  have  y  —  h-^  hK  By 
putting  positive  and  negative  small  values  of  h  in  this,  we 
easily  get  the  cusp  of  the  first  kind  which  is  formed  at  the 
upper  extremity  of  h, 

6.  "  To  find  the  cusps  of  the  curve  expressed  by 
y  =  a  +  x-\-  {x  —  hf.'' 

The  equation  is  satisfied  by  putting  x  =  h  and  y  =  a  +  h] 
consequently,  putting  a;  =  J  +  ^,  the  equation  becomes 
y  =  a-^h  +  h-\-  A*. 

Taking  the  differential  of  this,  regarding  h  as  being  the 
independent  variable,  we  have 

^y        ^       3,-1  ,     (l^y  3   ,-4 

£=l  +  -^h',     and    J=-j^Al 

By  putting  A  =  0  in  the  first  of  these  equations,  we  have 

consequently,  setting  off  a?  =  i  from  the  origin,  and  erecting 
the  perpendicular  y  =  a  +  J  to  the  axis  of  a?,  the  proposed 
cui've  will  touch  ?/  =  a  4-  J  at  its  upper  extremity,  and  it 
clearly  results  from  y  z=i  a  -\-  h  -\-  h  +  A  ,  that  by  putting 
y  =^  a  -\-  h  -\-  k^  we  shall  have  ^  =:  A  +  A  ;  which  clearly 
shows  that  A  must  be  positive,  since  the  denominator  of  the 
index  in  Ii*  is  even,  while  its  numerator  is  odd,  so  that  k  will 
be  imaginary  for  all  negative  values  of  A. 

It  is  hence  clear  that  the  proposed  curve  may  be  repre- 
sented by  the  figure,  such  that  o  being  the  origin  of  the 
rectangular  axes  ox  and  oy  of  x  and  y,  h  is  set  from  o  in  the 
direction  of  x  positive,  and  the  ordinate  h  +  a  drawn  through 


CUSPS,   OR  POINTS   OF  EEGRESSION.  203 


w 


0    b 

its  extremity  in  the  direction  of  y  positive ;  then,  YW"  being 
drawn  through  the  upper  extremity  Y  of  J  +  a  inclined  to 
the  axis  of  x  at  half  a  right  angle,  by  drawing  parallels  to 
y  at  the  distances  h  on  the  axis  of  a?,  and  setting  +  A*  on  the 
line  above  YW  and  —  h^  below  it,  both  being  set  off  from 
♦YW;   it  is   plain,  from   what  is   shown   at  p.   157,   since 

-^2  =  V^  is  shown,  at  p.  202,  to  be  of  a  contrary  sign  to 

the  ordinate  ±  h^  both  above  and  below  YW,  that  the 
curve  passing  through  the  extremities  of  the  ordinates  will 
give,  as  in  the  figure,  a  curve  whose  concavity  is  turned 
toward  its  diameter  YW;  consequently,  Y  can  not  be  a 
cusp,  but  it  must  be  what  is  called  a  limits  since  the  curve 
touches  the  ordinate  h  -\-  a  at  Y,  and  lies  to  the  right  of  the 
ordinate ;  since  it  is  plain  that  h  can  not  be  negative,  without 
making  the  ordinate  ±  h*  imaginary. 

6.  To  find  the  cusps  of  the  curve  whose  equation  is 

.  X  5 

2/  =  -  1  +  3  +  ^' - . 
At  the  origin,  or  when  a?  =  0,  we  have  x—0  and  y  =  —  1^ 
aud,  by  taking  the  differentials,  we  have  -y^  =  -  H-  ^  a?^, 

which,  at  the  origin,  gives  ^^  —  -  for  the  tangent  of  the 
angle  which  the  tangent  to  the  curve  at  the  origin  makes 


204  CUSPS,   OR  POINTS  OF  REGRESSION. 


/ 


with  tlie  axis  of  x ;  and  since  the  index  in  a?-  has  an  even 
denoniinator,  x  must  be  positive.  Hence  it  is  clear  that  the 
curve  has  a  cusp  of  the  first  kind  below  the  origin,  which 
may  be  represented  by  the  annexed  figure;  o  being  the 
oi'igin,  and  the  cusp  is  at  the  distance  1,  below  it. 

7.  "  To  find  the  cusps  of  the  curve  whose  equation  is 
y  =  a  +  a?  +  ^^"  +  cx'^-'' 

At  the  origin,  a?  =  0  and  y  =  a^  also  ~  =  1 ;  consequently, 

since  x  must  be  positive,  the  curve  clearly  has  a  cusp  of  the 
second  kind  above  the  origin  when  l  is  positive.  Hence  the 
cusp  may  be  represented  by  the  following  figure ;  in  which 
0  is  the  origin,  and  the  cusp  is  at  the  distance  a  above  it  (See 
Appendix,  p.  602,  &c.) 


SECTION  YIII. 

PLANE  AND  CURVE  SURFACES. 

(1.)  Plane  Surfaces. — 1.  A  plane  surface  may  be  repre- 
sented by  referring  it  to  three  rectangular  axes.  Thus,  let 
two  equal  lines  in  a  plane,  called  the  axes  of  x  and  y,  cut 
each  other  perpendicularly  at  the  point  0,  called  their  origin^ 
then  imagine  a  right  line,  called  ilie  axis  of  2,  to  be  drawn 
through  0  at  right  angles  to  the  plane  of  x  and  y,  or  to  that 
in  which  they  are  drawn. 

Hence,  suppose  a  plane  cuts  the  axis  of  z  at  the  distance 
c,  supposed  positive,  from  the  origin  O,  and  that  it  cuts  the 
planes  in  which  the  axes  of  z  and  a?,  and  those  of  z  and  y, 
lie  in  two  right  lines,  called  the  traces  of  the  plane.  Then 
from  any  point  in  the  trace  on  the  plane  s  and  ;»  draw  a  per- 
pendicular to  the  plane  a?,  y^  and  it  will  cut  the  axis  of  x  at 
the  distance  a?,  supposed  positive,  from  0,  the  origin  of  the 
co-ordinates ;  and  if  a  denotes  the  natural  tangent  (or  the 
tangent  to  radius  1)  of  the  angle  which  the  trace  makes  with 
the  axis  of  a?,  we  shall  clearly  have  ax  -^  c  for  the  length  of 
the  perpendicular  to  the  plane ;  a  being  positive  when  the 
perpendicular  is  greater  than  ^,  and  the  reverse.  And  from 
the  point  in  the  trace,  draw  a  right  line  parallel  to  the  trace 
on  the  plane  ^,  y^  and  draw  a  pei'pendicular,  2,  from  any 
point  in  this  line  to  the  plane  a-,  y ;  then,  y  being  the  dis- 
t-ance  of  this  point  from  the  plane  z,  x  and  h  the  natural 
tangent  of  the  angle  made  by  the  line  with  the  axis  of  y, 
we  shall,  as  before,  have  z  =^  c  -\-  ax  -\-  by  for  the  equation 


206         PLANE  AND  CURVE  SURFACES, 

of  the  plane ;  in  which  b  is  positive  when  z  is  greater  than 
the  preceding  perpendicular,  and  the  reverse. 

2.  It  is  easy  to  perceive  how  we  may  represent  the  equa- 
tions of  a  right  line  perpendicular  to  the  plana  Thus, 
imagine  planes  to  be  drawn  through  the  perpendicular  at 
right  angles  to  the  traces  z  =  ax  -\-  c^  and  z  =  hy  -\-  c] 
then,  from  well-known  principles  of  geometry,  the  common 
sections  of  these  planes,  and  the  planes  2,  x  and  s,  y,  will  be 
perpendicular  to  the  corresponding  traces. 

Hence,  from  what  is  shown  at  'p.  129,  we  shall  have 

X  11 

z  =  —  -  -h  c\      and    z  =  —  ^  +  c", 
a  0 

or  az  -\-  X  -^  c'  —  0,     and     J2  +  y  +  c"=  0, 

will  represent  the  equations  of  the  perpendiculars  to  the 

traces,  or  of  the  perpendicular  to  the  plane. 

3.  It  clearly  results  from  what  has  been  done,  that,  if  we 
please,  we  may  represent  the  equation  of  a  plane  by  the  more 

general  form,       Kx  +  By  +  C2  +  D  =  0 ; 

or,  if  it  passes  through  a  point  whose  co-ordinatesL  are  X,  Y, 

and  Z,  by    A  (X  -  a^)  +  B  (Y- y)  +  C  (Z  -  2)  =  0; 

and  ^  (Z^-  z)  -  (X'-  X)  =  0,     |-(Z'-  .)  -  (Y^-  y)  =  0, 

will  represent  the  equations  of  a  perpendicular  through  the 
point  X',  Y',  Z',  to  this  plane. 

(2.)  Curve  Surfaces. — 1.  Let  z  =f  {x,  y)  represent  the 
equation  of  a  curve  surface  referred  to  the  three  rectangular 
axes  of  X,  y,  and  s,  regarding  x  and  y  as  being  independent 
variables  ;  then,  if  Aa?'+  By'+  0/+  D  =  0  is  the  equation 
of  a  plane  referred  to  the  same  axes,  which  touches  the  curve 
surface  at  the  point  whose  co-ordinates  are  represented  by 
x\  y\  and  z\  it  is  plain,  if  the  partial  differential  coefficients 


PLANE   AND   CURVE   SURFACES.  207 

dz'  <l.z' 

■J-  and  -r— ,  of  tlie  surface  are  represented  by  f  and  ^,  that 

A  B 

we  must  liave  ^  =  —  --^,  and  q  =i  —  — -,  tlie  values  of  tlie 

7    /  ^7^/ 

partial  differential  coefficients  y-,  and  ~ ,  given  by  the  tan- 

A  B 

gent  plane.     Hence,  substituting  p  and  q  for  —  p-  and  —  p 

in  the  equation  of  the  plane,  reduced  to  the  form 
,  _        A    ,       ^    f  _R 

we  shall  have    z'  =  px'  +  qy' p  for  the  equation  of  the 

plane  which  touches  the  surface  at  the  point  {x\  y\  z'). 

Hence  we  have  Z  —  z— p(X  —  x)-]-q(J^—y)  for  the 
equation  of  a  plane  that  passes  through  the  point  (X , Y,  Z) 
without  the  surface,  and  touches  it  at  any  one  of  its  points 
(a?,  y,  z) ;  and  from  the  equations  of  the  perpendicular, 

--^(Z-^)+X'-a^^O,   and   _-?-(Z^-2)+Y'-y=.0, 

ioZ-z^-^{X-x)-^{Y-y)=p{yi-x)  +  q{Y-y\ 

^q\i^yq  p{Z'-z)+X!-x  =  0,  and  ^  (Z'- ^)  +  Y'-y  ==  0, 

for   the   equations   of  a   perpendicular  through   the  point 

(X',  Y',  Z')  to  the  tangent  plane. 

If  a,  5,  c   denote  the   angles   which  the   perpendicular, 

called  the  normal^  makes  with  the  axes  of  a?,  y,  and  2,  we 

shall  have 

X.'-x  ,      Y'-y        ,  Z'-z 

cos  a  =  — ^j^ — ,  cos  0  =  — ^r^-^,  and.  cos  c  =  — -^ — , 

in  which     N  =:  [(X'  -  xf  +  (Y'  -  yf  +  {Z'  =  zff ; 
consequently,  substituting   —p{Z'—z)   and    —  q  {Z' —  z) 


208  CHANGING   CO-ORDINATES. 

iroru  the  equations  of  the  normal,  for  X'  —  x  and  Y'  —  y, 
we  easily  get 


cos  a  =■ 


-P 


cos  h 


iMy  +  .f  +  i)^  -'■'-Vip'  +  f  +  iy 
■^^  '°' "  =  v(FT?TTy 

It  may  be  added,  if  z  is  not  explicitly  given  in  terms  of 
X  and  y,  but  implicitly  by  sucb  a  function  as  u  =0  =  sl 
function  of  a?,  y,  and  2 ;  tben,  by  taking  the  differential  co- 
efficients, we  sball  have 

du       du    dz       du      da  ^  ,     dii      du  ^ 

,  .  ,     .  da      du  ,  du      du 

whichgive     ^=-_^^,     and    5=--^^-. 

Hence,  we  must  substitute  these  values  of  p  and  q  for  them, 
in  the  preceding  equations.     (See  Young,  p.  163.) 

2.  There  are  one  or  two  transformations  of  co-ordinates 
that  are  often  useful  in  the  determination  of  the  forms  of 
curve  surfaces,  which  we  will  now  proceed  to  notice. 

1st  To  change  the  origin,  without  altering  the  directions 
of  the  co-ordinates. 

Let  a;,  y^  and  z  be  the  co-ordinates  of  any  point ;  then,  by 
putting  a  -h  a?',  5  +  y\  and  c  +  z\  for  a?,  y,  and  2,  the  co-ordi- 
nates will  be  changed  into  the  new  co-ordinates  x\  y\  and  z\ 
by  moving  the  origin  through  the  distances  (here  supposed 
positive)  a,  5,  and  c,  in  directions-  parallel  to  the  axes  of 
a?,  y,  and  z. 

2d.  To  transform  the  co-ordinates  of  a  point  when  referred 
to  two  rectangular  axes,  into  three  co-ordinates  referred  to 
three  rectangular  axes  having  the  same  origin. 


CHANGING   CO-ORDINATES,    ETC. 


209 


Let  O^'  and  Oy'  represent  tlie  given  rectangular  axes,  and 
P  a  point  in  their  plane  having  OP^  and  OP'^,  denoted  by  x' 
and  y\  for  its  co-ordinates.  Through  the  origin  0  of  the  co- 
ordinates let  Ox  be  drawn,  making  the  angle  xOx'  =  0  with 
Ox\  and  let  the  plane  of  Ox  and  O.^;'  make  the  angle  O  with 
the  plane  of  the  lines  Ox'  and  Oy'.  Then,  draw  Oy  in  the 
plane  of  Ox'  and  Ox  at  right  angles  to  Ox,,  and  Oz  at  right 
angles  to  the  plane  of  Ox  and  Ox'^  or  Oy.  Join  OP,  then 
a?,  y,  s,  the  co-ordinates  of  P  when  referred  to  the  rectangular 
axes  of  0.»,  Oy,  and  0.2,  are  clearly  the  projections  of  the 
distance  OP  on  the  axes,  which  are  clearly  the  sums  of  the 
projections  of  OP^^  and  PP'^,  or  OP^'  and  OP',  on  the  same 
axes.  Because  O^'  is  perpendicular  to  the  lines  Oy'  and  0,a:, 
it  is  clearly  perpendicular  to  their  plane,  and  in  like  manner 
Oy  is  perpendicular  to  the  plane  of  the  lines  Ox  and  O^; 
consequently,  the  angle  made  by  these  pianos  with  each 


210  CHANGING  CO-ORDINATES,   ETC. 

other  equals  the  angle  yOx'  =  -  —  0,  by  representing  the 

right  angle  xOy  by  ^,  since  <f>  =  the  angle  xO:c'.     Since 

OP'  cos  0  =  a;'  cos  <^  =  the  projection  of  x'  on  the  axis  of  .r, 

and  that  OP''  cos  I  ^  —  </>)  cos  0  =  y'  sin  (^  cos  Q  equals  that 

of  y'  on  a?,  we  shall  have  x=^x'  cos  (p  -\-  y'  sin  0  cos  0,  and 
in  like  manner 

y  =  x'  cos  (o  ~  *A)  +  y'  ^^s  ("""  ~~  ^)  ^^s  ^ 
=  x'  sin  (j)—  y'  cos  «^  cos  0, 
and         ^  =  y'  sin  0. 

Hence,  if  we  substitute  these  values  of  x,  y,  and  z,  in  the 
equation  of  any  surface,  the  resulting  equation  in  x'  and  y' 
will  give  the  equation  of  its  section  by  the  plane  of  x'  and 
y\  through  the  origin  of  the  co-ordinates,  and  thereby  give 
us  a  clearer  view  of  the  nature  of  the  surface. 

Thus,  if  we  take  a?^  -f-  y"  +  2^  =  r^,  the  equation  of  the  sur- 
face of  a  sphere,  and  make  the  preceding  substitutions,  we 
shall  have 

x'^  (cos^  (p  +  sin^  4>)  -f  y'^  cos^  d  (cos^  0  +  sin^  «/>)  +  y'^  siu^  6 

consequently,  the  section  of  a  sphere  by  a  plane  through  its 
center  is  a  circle  whose  radius  equals  the  radius  of  the  sphere. 

If  x'  cos  <f>  -{-  y'  sin  (p  cos  ^  +  a,  a?'  sin  <p  —  y^  cos  0  cos  0  -\-h, 
and  y'  sin  ^  +  c,  are  put  for  a?,  y,  and  2,  in  the  equation  of  a 
surface,  we  shall  have  the  equation  of  a  section  of  the  sui*- 
face,  when  the  origin  of  the  co-ordinates  is  changed,  without 
altering  their  directions. 

By  making  these  substitutions  in  the  equation  of  the  sur- 
face of  the  sphere,  we  readily  get 


REPRESENTIN-G   CYLTNDRIC   SURFACES,   ETC.  211 

ix'  +  a  cos  </)  -f  J  sin </)) ^  +  (2/'  +  a  sin  0  cos  Q—h  cos  <^  cos  0  +  c  sin  Q^f 
=z  7^'  —  {a  sin  <^  sin  ^  —  J  cos  (/>  sin  0  —  c  cos  oy^ 

for  the  equation  of  the  section  of  the  surface  by  the  plane 
of  x'  and  y\  which  is  clearly  the  equation  of  a  circle  whose 
radius  is  the  square  root  of  the  right  member  of  the  equa- 
tion. 

If  a  —  0,  J  ==  0,  and  0  =  0,  the  equation  reduces  to 

X''  +  y'2  ^  ^2  _  ^.2^ 

the  cutting  plane  being  at  the  distance  c  from  the  center  of 
the  sphere.      If  <?  =  r^  the  section  becomes 

consequently,  we  must  have  x^  -=0  and  y'  =  0,  and  the 
cutting  plane  becomes  a  tangent  to  the  surface  of  the  sphere, 
and  is  clearly  at  right  angles  to  the  radius  drawn  to  the 
point  of  contact.  If  c  is  greater  than  /•,  we  shall  have 
^/2  _|.  y'l  _  ^.2  _  ^,2^  ^  negative  result,  which  is  impossible ; 
consequently,  the  plane  of  x'  and  y'  does  neither  cut  nor 
touch  the  spheric  surface.  If  c  =  0,  the  equation  becomes 
a?'-  +  y''^  —  'p^^  which  is  called  the  equation  of  a  great  circle 
of  the  sphere,  while  that  of  x'^  +  y''^  =  r^  —  c-  is  called  that 
of  a  small  circle. 

(8.)  We  will  now  proceed  to  show  how  to  represent 
cylindrical,  conical,  and  surfaces  of  revolution,  &c.,  accord- 
ing to  the  methods  of  Monge. 

1.  Let  x  —  az-\-a'  and  y  ^hz  -{-V  represent  the  projec- 
tions of  a  right  line  in  space,  on  the  planes  of  the  rectangular 
axes  of  (a",  2)  and  (y,  s),  which  moves  parallel  to  itself,  and 
during  its  motion  continually  passes  through  the  common 
section  of  two  surfaces,  represented  by  the  equations 
F  (a?,  y,  s)  =  0  andy  {x^  y^  z)  —  {)\  then,  the  generated  sur- 
face is  said  to  be  of  a  cylindrical  form^  the  moving  right 


212  REPRESENTING   CYLINDRIO  SURFACES,   ETC. 

line  being  called  its  generatrix^  while  tlie  intersecting  sur- 
faces are  called  its  directrix. 

Because  the  generatrix  is  constantly  parallel  to  itself,  it  is 
manifest  that  a  and  h  must  be  constant  or  invariable,  while 
a'  and  V  will  vary.  But  if  we  eliminate  a?,  y,  z  from  the 
four  equations,  which  must  be  constantly  coexistent,  we 
shall  get  an  equation  that  may  be  expressed  by  the  form 
V  =  <})  (a/)  =  a  function  of  a',  or  since  a'  =:  x  —  as  and 
h'  ^y  —  bz^  we  have  y  —  hz  =  (f)  {x  —  az)  for  the  general 
form  of  equations  of  cylindrical  surfaces. 

Differentiating  the  equation  separately,  by  regarding  z  as 

a  function  of  a?,  and  then  by  regarding  s  as  a  function  of  y, 

we  shall  have 

—  lpz=  (1  —  ap)  (f)'  (x  —  az) 

and  1  —  J}q=  ~  aq(f)'  (x  —  az); 

in  which  0'  {x  —  az)  is  put  for  — ^^ -,  when  0  (a?  —  az) 

is  regarded  as  being  a  function  of  z.  Eliminating  (f)'  {x  —  az) 
from  the  equations,  we  get  t-^^'IT  ~ '>  ^^  '^^^  equiva- 
lent ap+hq=z\^  which  is  called  the  general  differential 
equation^  or,  more  properly,  that  of  partial  differential  co- 
efficients, of  cylindrical  surfaces.  The  same  equation  results 
immediately  from  Z  —  s  =  j9  (X  —  a?)  +  <?  (Y  —  y),  the  gen- 
eral equation  of  a  tangent  plane  to  cylindrical  surfaces.  For 
the  equations 

x-=:az  -{•  a^    and     y  ^=^hz  •{-  h\ 

give     X  —  a?  =  a  (Z  —  s)    and    Y  —  y  =  h  {Z—z\ 
which,  being  substituted  in  the  tangent  plane  and  the  com- 
mon factor  2i  —  z  rejected,  becomes  aj9  +  J^  =  1 ;  the  same 
as  before.     If,  as  at  p.  208,  -w  =  0  is  an  implicit  function  of 
a?,  ?/,  and  z,  we  shall  have 


REPRESENTING  CYLINDRIC   SURFACES,    ETC.  213 

du        die  T        __       du    ^    da 

■^  ~~       dx    '    dz  ^  ~~       dy    '    dz 

which,  substituted  in  the  preceding  equation,  give 

die        ,  da        du        . 
ax  dy        dz 

which  is,  apparently,  of  a  more  general  form  than  the  pre- 
ceding equation. 

2.  To  determine  the  general  equation  of  conical  surfaces, 
we  may  proceed  in  much  the  same  way  as  before,  by  taking 
x  —  x'  =  a{z—z')  and  y  —  y'  =1)  {z  —  z')  for  the  equa- 
tions of  the  right  line  which  constantly  passes  through  its 
vertex  {x'^  y\  z')  supposed  fixed,  and  by  passing  during  its 
motion  continually  through  the  directrix  F  (a?,  y,  z)  =  0  and 
y  (a?,  y,  z)  =:  0,  generates  the  conical  surface. 

x\  y\  and  z'  being  supposed  to  be  known,  if  we  eliminate 
£c,  y,  and  z  from  the  four  preceding  equations,  we  shall,  as 

before,  get '-—  =  (p  I '- 7 1  for  the  required  equation. 

z  —  z  \  z  —  Z  I 

Eegarding  the  right  member  of  this  equation  as  being  a 
function  of  s,  and  eliminating  the  function  as  in  the  case  of 
cylindrical  surfaces,  we  readily  get 

{y-y')P  ^  z-z'  -{x-x')p 

^-^' -ky-y')q.  {x-x')q       ' 

or  its  equivalent  z  —  z'  :=p(x  —  x')  -{-  q{y  —  y%  which  is 
the  equation  of  the  partial  differential  coefficients  of  tlie 
general  equation  of  conical  surfaces,;  which  is  clearly  the 
general  equation  of  the  tangent  plane  to  the  conical  surface, 
as  it  ought  to  be.  If  -w  =  0  is  an  implicit  function  of  a?,  y, 
and  2,  we  shall  have 

du    ^    du  ,        du        du 

^  ~~       dx    '    dz  ^  ~       dy    '    dz'' 


214  REPRESENTING  CYLINDRIC  SURFACES,   ETC. 

which  reduce  the  preceding  equation  to 

Thus,  if  for  w  =  0  we  take  xyz  —  3y-£c  —  20  =  0,  we  shall  get 
yz  —  3y*,  xz  —  6a?y,  and  xy^  for  the  values  of  y-,  -j-,  and  -y^  ; 

or,  accenting  the  letters  in  these  expressions  to  signify  that 
they  correspond  to  the  line  of  contact  of  a  conical  surface 
with  the  proposed  surface,  the  above  equation,  by  substitu- 
tion, becomes 

{x-x')  (y'z'-Zy")  +  {y-y')  {x'z'-Qx'y')  +  {z-z')  x'y'^0', 
or,  since  x'y'z'  —  %y'^x'  =  20,  it  becomes 

X  (yz'  -  Sy'')  +  y  {x'z'  -  ^x'y')  +  zx'y'  =  60  ; 
which  is  called  the  equation  of  the  conical  surface,  which 
envelops  the  proposed  surface,  (x,  y,  z)  being  the  vertex  of 
the  conical  surface,  or  of  the  enveloping  surface.  The 
equation  being  of  the  second  degree  in  x\  y\  and  z\  shows 
that  the  line  of  contact  of  a  conical  surface,  having  its 
vertex  at  the  point  {x,  y,  s),  with  the  proposed  surface,  is  in 
a  surface  of  the  second  degree.  It  is  hence  easy  to  perceive, 
that  if  a  conical  surface  envelops  a  surface  of  the  wth 
degree,  that  the  line  of  contact  will  be  in  a  surface  of  the 
[in  —  l)th  degree.  (See  "  Application  de  1' Analyse  a  la 
Geometric,"  by  Monge,  pp.  14  and  15;  also,  see  Young, 
pp.  170  and  171.) 

3.  To  find  the  general  form  of  the  equations  of  surfaces 
of  revolution. 

Let  X  =  az-^  a\  and  y  =  hz-\-  V ,  represent  the  equations 
of  the  axis  of  revolution ;  then,  from  what  is  shown  at  p.  20t5, 
2  +  ««  -f  2»y  =  c  is  the  equation  of  a  plane  perpendicular  to 
the  axis. 


EEPKESENTING   CYLINDEIC   SUKFACES,    ETC.  215 

Because  tlie  perpendicular  plane  cuts  the  surface  in  the 
circumference  of  a  circle  whose  center  is  in  the  axis,  by 
representing  the  co-ordinates  of  the  center  of  a  sphere,  of 
which  the  circle  is  a  section  by  the  perpendicular  plane,  by 
a',  h\  and  0 ;  we  shall  have  (x  —  a'f  +  (y  —  h'f  -\-  z'^  —  r'^ 
for  the  equation  of  its  surface,  in  which  we  may  suppose 
a'  and  h'  to  be  constant,  while  a?,  y,  2,  and  the  radius  r,  are 
variable. 

If  we  substitute  the  values  of  {x  —  a'f  and  (?/  —  Vf  from 
the  equations  of  the  axis,  in  the  preceding  equation,  it  be- 
comes (a^  +  ^^  +  1)  2^  —  T^i  in  which  a  and  h  are  constant,  or 
invariable ;  and  substituting  ax  and  hj  from  the  equations  of 
the  axis  in  that  of  the  perpendicular  plane,  it  becomes 
(a^  +  J2  +  1)  2  rf  a^'  +  W  =  G. 

It  is  hence  manifest,  from  a  comparison  of  these  equations, 
that  we  may  assume 

z-^ax-\-'by=<t>l{x-  aj  -\-  {y  -  IJ  +  ^"] 

for  the  general  equation  of  surfaces  of  revolution. 
li  a'  =  0  and   h'  =  0,  the  equation  is  reduced  to 

3-\-ax  +  hy  =  ({){x''  +  y''  +  b% 
which,  when  a  =  0  and  h  =  0,  or  when  the  axis  of  revolu- 
tion coincides  with  that  of  z,  becomes  z  =  (f)  (x^  +  ^^  +  z^)^ 
or,  more  simply,  we  shall  have  z  ='  t/>  (x^  +  y^). 

If  we  regard  the  right  member  of  the  general  equation  of 
surfaces  of  revolution  as  being  a  function  of  z^  we  shall, 
from  the  elimination  of  the  arbitrary  function,  as  heretofore, 

.  , ,  , .  p  +  a      X  —  a'  -{-  pz 

ffet  the  equation ^  = r-. — ^— , 

^  ^  q-{-h       y-h'+qz' 

or  its  equivalent, 

(y'~  h'—lz)  p—  {x—a'—az)  q-\-a  {y~J/)  -  h  {x—a')  =  0; 


216  BEPRESENTING  CYLINDRIC  SURFACES,   ETC. 

which,   when    z    is    the    axis    of   revolution,    reduces    to 

yp  —  xq  =  0. 

The  same  equations  among  the  preceding  partial  differen- 
tial coefficients  may  readily  be  obtained  from  the  equations, 

x  —  x'-\-p{z  —  z')  =  0,     and     y  —  y'  +  q  {z  — z)  =  0, 
of  the  normal  to  any  point  {x\  y\  z')  of  the  surface  of  revo- 
lution, as  is  evident  from  the  consideration  that  the  normal 
must  pass  through  the  axis  of  revolution,  whose  equations 
must  clearly  be  coexistent  with  those  of  the  normal. 

Hence,  eliminating  x  and  y  from  the  equations  of  the  nor- 
mal by  means  of  the  equations  of  the  axis,  they  will  be 
reduced  to 

az  +  a'  —  x'  +jpz  —  pz'  =  (a  +p)  z  -\-  a'  —  x'  —  pz'  =  0, 

and  {h  +  q)z  -}-  y—y'  —  qz'  =  0; 

consequently,  eliminating  z  from  these  equations,  we  shall 

,  a  +  p        x'  —  a'  ■\-  pz' 

have  1 —  =  —, 77 ^—:, 

h  +  q        y'  -  b'  -^  qz" 

which  agrees  with  the  equation  at  p.  215,  when  we  use  a?,  y, 
and  z  for  x\  y\  and  s',  as  at  the  place  which  has  been  cited ; 
hence,  all  the  preceding  results  will  be  obtained,  as  alcove. 

To  illustrate  what  has  been  done,  let  a;  =  as  +  a',  and 
y  =  Js  -f-  h'^  represent  the  equations  of  a  right  line  revolving 
around  an  axis  parallel  to  the  axis  of  s,  to  find  the  nature  of 
the  surface  of  revolution  described  by  it. 

From  s  =  t/>  (ar  -f  ?/*),  we  clearly  get  a?^  +  y^  =  t/»'s  =  a 
function  of  z ;  which  clearly  becomes 

x'  +  f  =  {az^-  aj  +  {hz  +  bj, 
from  the  substitution  of  the  values  of  x  and  y  from  the  equa- 
tions of  the  revolving  line. 

To  determine  the  nature  of  the  surface  more  fully,  we 


KEPHESENTING   CYLINDRIC   SURFACES,    ETC.  217 

shall  find  the  nature  of  its  section  by  a  plane  through  its 
axis.     Thus,  substituting  the  values  of  a?,  y,  and  ^,  from 

7T 

p.  210,  since  0  =  -  =  90°,  we  have  sin  0  ==:  1  and  cos  0  ==  0 ; 
and  thence,  get    x  ^=^  x'  cos  0,     y  ==  a?'  sin  0,    and    z  =  y'. 
Hence,  from  the  substitution  of  these  values  in  the  preceding 
equation,  since  sin^  </>  +  cos^  0  =  1,  we  shall  have 
x'^  =  {ay  +  ay  +  {hy'  +  hj 

for  the  equation  of  the  section  of  the  surface  by  a  plane 
through  the  axis  of  s,  which  is  perpendicular  to  the  plane 
of  X  and  y.  Developing  the  right  member  of  the  equation, 
we  have 

x''  =  {a'  -}-  ¥)  y'^  +  2  {aa'  +  W)  y'  +  a""  +  l'\ 
or  its  equivalent, 

,.  =  (a»  +  V)  (y'  +  ^t|^)  -  ^^J^'  +  «"  +  J'^ 

which,  by  representing  x'  and  2/'  H 2~rT2~  ^7  ^  ^^<i  Y, 

a  ~\~  0 

is  readily  reducible  to 

6f    "f"  0 


the  equation  of  an  hyperbola,  having  Y  =  ±  X  -^  i^\a^  +  V) 
for  the  equation  of  its  asymptotes. 

Hence  it  is  clear  that  the  equation  above  found  is  that  of 
an  hyperholoid  of  revolution  of  one  sheet,  (See  page  20 
of  Monge's  work.) 

4.  A  given  curve  surface  revolves  roimd  a  given  straight 
line,  to  find  the  surface  which  touches  and  envelops  the 
moving  surface  in  every  position. 

The  required  surface  must  clearly  be  a  surface  of  revolu- 
tion roond  the  given  straight  line ;  consequently,  the  curve 

10 


218  BEP RESENTING  CYLINDRIC  SURFACES,   ETC. 

of  contact  of  the  sought  surface  and  the  revolving  surface 
in  its  first  position  is  evidently  a  curve  whose  revolution 
round  the  given  straight  line  will  generate  the  required 
surface.  It  is  hence  clear  that  this  question  is  reducible  to 
that  given  on  page  214,  and  that  jt?,  q^  in  the  revolving  curve, 
must  be  the  same  as  in  the  revolving  surface  in  its  first 
position,  and  that  they  must  satisfy  the  equation  of  condition 
as  there  found. 

Thus,  let  the  revolving  surface  be  that  of  a  spheroid, 
having  x^  -\- y^  -\- n- z-  =  ni?  for  the  equation  of  its  surface  ; 
and  supposing  it  to  revolve  round  one  of  its  diameters 
having  a?  =  as  and  y  =  Js  for  its  equations,  when  referred 
to  its  principal  diameters;  then  from  the  equation  of  the 

surface  we  shall  get   jr?  = —    and  <7  = ^. 

Because  a',  5',  each  equal  naught  in  this  example,  the 
equation  at  page  215  becomes 

{y  —  hz)  p  —  {x  —  az)  q  +  ay  —  hx  =  0] 

which  the  substitution  of  the  preceding  values  of  p  and  q 
reduce  to  ay  —  Ja?  =  0.  Hence  the  equations  of  the  gen- 
erating curve  of  the  envelope  are  expressed  by 

ix?  +  if  +  n^  z^  =^  771^,     and    ay  =^  hx] 

and  because  the  described  surface  is  a  surface  of  revolution, 
we  must  also  have  (see  p.  215) 

ax -^  hy  -\- z  =  c,     and     or  -\-  f  +  z'  =  r^. 

From  the  first  and  fourth  of  these  equations,  we  have 

and  since  the  second  and  third  give 

ay—hx  =  0     and     ax  -\- hy  =  o  —  z, 


DEVELOPABLE   SURFACES.  219 

we  have,  bj  taking  tlie  sum  of  their  squares, 

which,  from  x"  -\-  y'^  =  r^  —  s^,  is  reduced  to 
(a^  +  5^)  if-s")  =  {c-z)\ 
Hence,  from  the  substitution  of  the  value  of  z  in  this 
equation,  we  have 

(ct^  +  hY  {nh'"'  -  m?)  =  [g^  {jv"  -  1)  -  \/{m^  -  7^)]% 
for  a  conditional  "equation  among  the  four  preceding  equa- 
tions that  involve  a?,  y,  and  z. 

Since  ?''  =  a?^+  yM-  ^^  and  c  =  ax-{-hy  -{-  2,  the  preceding 
equation  is  equivalent  to 

[(s  +  aa?  -I-  hj)  |/(?i2  —1)  —  ^/(m^—  x^  —  f  —  2^)]% 
which  is  the  equation  of  the  sought  or  enveloping  surface ; 
agreeing  with  Mr.  Young's  result,  at  p.  176  of  his  work. 

5.  When  a  surface  is  such,  that  it  can  be  conceived  to  be 
spread  out  on  a  plane  without  being  torn  or  rumpled,  it  is 
called  a  (ler.elopahle  surface. 

It  is  clear  that  a  developable  surface  may  be  expressed  by 
means  of  its  tangent  plane  as  follows : 

Thus,  let  Z  —  2  =  jp  (X  —  a?)  +  q  (^  —  y)  represent  the 
equation  of  its  tangent  plane,  which  is  easily  put  under  the 
equivalent  form  Z  =  jpX  +  qY  -\-  z  —  px  —  qy:  in  which 
2,  x^  y^  are  the  co-ordinates  of  the  point  of  contact  of  the 
plane  with  the  surface ;  while  Z,  X,  Y,  are  the  co-ordinates 
of  any  other  point  of  the  plane. 

Supposing />  and  q  to  be  constant,  or  their  total  differentials 

dz 
to  equal  naught,  while  a?,  y,  and  z  are  changed,  since  -r-  =z  p 

dz 
and  —  z=  q^  we  easily  get    in   differential  coefficients,  the 


220  DEVELOPABLE  SURFACES. 

general  equation  of  developable  surfaces.    Thus,  representing 

dp  _(r-2    dq  _d^z          ,     ^M.  —  ^  ^'^g    _    <^-g 

dx  ~  d?'  d^  ~  dif'              dy  ~~  dx'  dxdy  ~  d/ydx^ 

severally,  by  r,  t^  and  5,  by  putting  the  differentials  of  p 
and  q  equal  to  naught,  we  have 

'^^dx  +  ^dy=0     and    "^  dx  ^  ^  dy  =  0,         ' 
dx  dy    "^  dx  dy    ^ 


or 


rdx  4-  sdy  =  0        and  sdx  +  tdy  =  0, 


-  .  ,      .  di/  r  ,  dy  s 

which  give  ^=-J  ^^d  d-x=--i' 

Equating  these  values  of  ;t^  ?  we  have 

-  =  -,  or  rt  —  ^=0. 
s       t 

which  is  equivalent  to 

d^       d^  _  /  d^2  Y_ 

dx^       dy^        \dxdy)  ~    ' 

for  the  equation  of  partial  differential  coefficients  of  the  sec- 
ond order  of  developable  surfaces. 

Eesuming,  rdx  +  sdy  =  0  and  sdx  +  tdy  =  0,  and  multi- 
plying the  first  by  dx  and  the  second  by  dy,  by  adding  the 
products  we  have 

rdx^  +  2sdxdy  +  tdy'^  =  0, 

which  is  called  by  Monge  (at  p.  82  of  his  work),  the  charao- 
teristic  of  developable  surfaces. 

Because  dz  ^z^jpdx  +  qdy,  we  shall  clearly  have 
^-z  =  rclji?-  +  2sdxdy  +  tdy^, 

consequently,  since  the  right  member  of  the  equation  equals 
naught,  we  shall  have  d-z  ==  0  in  case  of  a  plane ;  that  is, 
cZ^^  =  0  is  the  characteristic  of  developable  surfaces. 


DEVELOPABLE  SURFACES.  221 

Since  d^z  ==  0  belongs  to  a  plane,  and  that  tlie  contact  of 
the  tangent  plane  with  the  surface  is  clearly  a  line,  it  is  evi- 
dent that  all  the  points  of  the  line  may  be  regarded  (see 
Monge,  p.  82)  as  constituting  a  plane  line. 

Because  'rt  ^=  r,  we  shall  have  ,9  =  Vrl,  which,  being  put 
for  s  in  the  characteristic,  reduces  it  to 

rdx^  +  2dxdf/  Vrt-{-  tdif  =  {djc^/r  -f  di/^{f  —  0, 

whose  square  root  gives 

dxVr  +  dy^t  =  0,    ov    ^=-/'-,    or    J=-/^; 

these,  when  the  surface  is  developable,  are 

dy  _       r         ^     dy  _       s 
dx  s  dx~       t 

as  before  shown. 

dy             V  s 

Because  -p=: ,or  —  -=  the  tangent  of  the  angle 

which  the  projection  of  the  line  of  contact  on  the  plane  a?,  y, 
makes  with  the  axis  of  x,  it  is  clear,  from  what  has  been 
done,  that  the  line  of  contact  must  be  a  right  line ;  which 
may  be  regarded  as  being  the  generatrvx  of  the  developable 
surface. 

Hence  the  developable  surface  (by  Monge  called  the 
envelope  of  the  infiniteshnal  tangent  plane)  may  be  consid-  • 
ered  as  composed  of  plane  elements  of  unlimited  lengths  and 
of  infinitesimal,  breadths,  which  successively  cut  each  other  in 
right  lines.  Hence  the  first  of  these  elementary  planes  may 
be  turned  about  its  line  of  common  section  with  the  second, 
until  its  pLane  is  brought  into  the  same  plane  with  it ;  and  in 
like  manner  the  plane  thus  formed,  may  be  turned  about  the 
line  of  common  section  of  the  second  and  third  elements, 


222  developable:  surfaces. 

until  it  is  brouglit  into  the  plane  of  the  third  element ;  and 
so  on  to  any  extent  that  may  be  required.  It  is  hence 
evident  that  a  developable  surface  may  be  spread  out  on  a 
plane  without  being  torn  or  rumpled. 

Because  (see  pp.  212  and  213)  the  equations  of  cylindrical 
and  conical  surfaces  are  represented  by 

op  -\-  hq  =  1,     and     z'  ^=px'  -\-  qy'  -\-  z  —px  —  qy, 

it  clearly  follows,  from  what  has  been  done,  that  they  are 
developable  surfaces ;  since  they  evidently  come  under  the 
form  z  =pX  +  ^Y  -\-  z—px  —  qy,  in  which  the  differentials 
of  p  and  q  are  put  equal  to  naught,  X,  Y,  Z  are  constant, 
whUe  a?,  3/,  z  are  variable. 

Remarks. — If  we  assume  z  =  x^  (a)  +  yip  (a)  +  a  to  rep- 
resent the  equation  of  a  plane,  in  which  </>  (a)  and  i>  (a)  repre- 
sent any  arbitrary  functions  of  a ;  then,  by  putting  the  first 
and  second  differential  coefficients  taken  with  reference  to  a 
equal  to  naught,  we  shall  have  the  forms  (see  Monge,  p.  85) 

a?0'  (a)  +  yxj)'  (a)  +  1  =  0     and     a?</>"  (a)  +  y^''  (a)  =  0. 

Young  (at  p.  208  of  his  "  Differential  Calculus")  says,  that 
for  a,  in  the  equation  of  the  plane,  the  function  /" (a)  ought 
to  be  used,  since  Mongers  form  excludes  those  forms  com- 
prehended in  the  form  z  =  X(f>  (a)  +  yip  (a)  +  c ;  but  this 
objection  is  clearly  invalid,  since,  if  we  please,  we  may  for  z 
put  s  —  <?,  and  omit  a  to  suit  the  case,  and  we  have  Mr. 
Young's  form. 

It  is  evident  that  tlie  equation  in  x,  y^  and  s,  resulting 
from  the  elimination  of  a  from 

z  =  x(p  (x)  4-  yi>  («)  H-  a     and      xcp'  (a)  +  yxp'  (a)  +  1  =  0, 

represents  a  developable  surface.  For  the  first  equation  in 
virtue  of  the  second,  gives 


DEVELOPABLE   SURFACES.  223 

^  =  ^  rrr  0 (a),     and     -^  =  q  =  ^{a)- 

consequently,  since  p  and  q  are  functions  of  a,  we  shall  evi- 
dently liave  p  =  6{q)  =  SL  function  of  q.     Hence,  we  shall 

have  ^  =  ^  =  ^  X  '^^  =  se'is), 

ay  ax         dq  ^^' 

consequently,  eliminating  Q'  {q)  from  these  equations,  we  get 
rt  —  s^  =  0,  the  equation  of  developable  surfaces,  and,  of 
course,  the  assumed  equations  jointly  represent  a  developable 
surface. 

Kegarding  a  as  being  an  arbitrary  constant,  that  ought  to 
be  retained  in  the  equations,  it  is  clear  that  the  equations 
may  both  be  regarded  as  being  functions  of  the  characteristic, 
since  the  position  of  the  characteristic  clearly  depends  on  a. 

Hence,  the  first  equation  being  that  of  a  plane,  and  the 
second  that  of  a  right  line  on  the  plane  a?,  ?/,  it  is  manifest 
that  the  characteristic  must  be  a  right  line,  which  is  the  same 
as  the  generatrix  of  the  surface.     (See  Monge,  p.  85.) 

If  we  eliminate  a  from  the  equations 

z  =  x(}>  (a)  +  yi>  (a)  +  a, 

x(}>' (a) -{- yrp' (a)  +  1  =  0, 

and  x(p''  (a)  -f  yf' (a)  =  0, 

we  shall  clearly  get  two  equations  in  terms  of  a?,  y,  and  s, 
which  will  clearly  be  the  equations  of  the  line  in  which  the 
intersections  of  the  successive  characteristics  must  lie,  which 
must  evidently  be  on  the  developable  surface  ;  this  line  being 
called  by  Monge,  the  edge  of  regression  of  the  envelope^  or 
developable  surface.     (See  Monge,  p.  85,  and  Young,  p.  212.) 


224  ILLUSTRATIONS. 

We  will  illustrate  what  has  been  done  by  one  or  two 
simple  examples. 

1st  Let  z  =  xi^  -{-  ya  +  h  represent  the  variable  plane,  to 
find  the  developable  surface  and  its  edge  of  regression. 

Here,  by  taking  the  differential  coefficients  relatively 
to  a,  we  have  the  remaining  two  equations  expressed  by 
2xa  -{-  y  =  0,  and  2x  =  0.  Hence,  eliminating  a  from  the 
first  of  these  and  the  proposed  equation,  we  get 

4.x  (0  -  5)  +  f  =  0, 

for  the  equation  of  the  envelope ;  which  will  be  found  to  be 
a  developable  surfaca  If  we  eliminate  a  from  the  three 
equations,  since  2a?  =  0,  we  shall  get  s  =  5,  a  point  in  the 
axis  of  Sj  for  the  edge  of  regression. 

2d.  Let  the  variable  plane  he  2=  xa?  +  ya^  +  a ;  then,  the 
other  equations  are  Zxd?  +  2ya  +  1  =  0,  and  ^xa  -j-  y  =  0. 

Solving  the  second  of  these  equations  by  quadratics,  we 

get  a  =  —  —  ±  ^—~: ;   which,  substituted  for  a  in 

the  proposed  equation,  will  give  the  envelope  or  developable 

surface. 

Also,  eliminating  a  from  the  three  equations,  since  the 

V  1 

third  gives  a  =  —  ^,  we  have  Sx  =  y^  and  y2=  —  -^  the 

first  being  that  of  a  parabola  on  the  plane  a?,  ?/,  and  the 
second  that  of  an  hyperbola  on  the  plane  y,  2,  for  the  equa- 
tions of  the  edge  of  regression. 

6.  By  a  twisted  surface  we  mean  one  described  by  a  right 
line  which  is  continually  changing  the  plane  of  its  motion. 

To  represent  such  a  surface,  we  shall  suppose  x  =  az  -\-  a' 
and  2/  =  Js  +  2»'  to  be  the  equations  of  the  generatrix,  which 
we  shall  suppose  to  be  continually  moving  along  three  given 


TWISTED  SURFACES.  225 

curves  of  double  curvature  (or  which  do  not  lie  wholly  in 
the  same  plane)  as  directrices ;  then,  each  of  these  curves 
will  be  expressed  by  two  equations,  when  projected  on  the 
rectangular  planes  of  x,  ^,  and  <•/,  z. 

Hence,  if  we  eliminate  a?,  y,  and  s,  from  the  equations  of 
the  generatrix  by  the  equations  of  any  one  of  the  directrices, 
we  shall  have  an  equation  involving  a,  J,  a' ^  and  V^  as  un- 
known quantities  ;  consequently,  if  we  eliminate  a?,  y,  and  ^, 
in  like  manner,  from  the  equations  of  the  generatrix  by  the 
equations  of  the  remaining  directrices,  we  shall  have  two 
more  equations,  each  involving  a,  J,  a',  and  5',  as  unknown 
quantities.  Hence,  from  the  solution  of  the  three  equations 
thus  found,  any  one  of  the  quantities  a,  5,  a\  and  h'^  as  a, 
may  be  supposed  to  be  taken  for  the  independent  variable, 
and  each  of  the  others  to  be  a  function  of  it.  Thus,  for 
a',  J,  and  h\  we  may  put  ^  {a\  (p  (a),  and  6  {a) ;  which  reduce 
the  equations  of  the  generatrix  to  the  forms 

X  =  as  +  'ip  (a),    and    y  =  £(}){a)  +  0  (a), 

in  which  V',  0,  and  0,  that  precede  a,  are  used  to  denote  any 
arbitrary  functions  of  it ;  so  that  if  ip  (a)  is  assumed  to  equal 
naught,  </>  {a)  —  a^,  and  6  {a)  =  a^,  our  equations  will  become 
X  —  nz  and  y  =  a%  +  a^,  which,  by  eliminating  a  from  the 
second  by  the  first,  give  ys^  =  qi?^"  +  x^^  for  the  equation  of 
a  twisted  surface. 

Generally,  if  a  is  found  from  one  of  the  equations, 
X  =  az  +  i/^  (a)  and  y  —  z(f){a)  +  6  (a),  and  its  value  substi- 
tuted in  the  other,  z  will  become  a  function  of  x  and  y,  or 
we  shall  have  the  equation  of  a  surface,  since  ^  is  a  function 
of  X  and  y ;  consequently,  as  heretofore,  we  shall  have 
dz  =  pds0  -h  qdy^  in  which  x  and  y  are  the  independent 
variables. 

10* 


226  TWISTED  SURFACEa 

By  taking  the  differentials  of  x  =  as  -{-  i>  (a)  and 
y  =  e:){a)  -f  0{a),  supposing  a  to  be  constant,  we  shall 
have  dx  =  adz  and  dt/  =  (f)  (a)  dz  ;  which  show  that,  in  taking 
tlie  differential  oi  dz  =pdx-[-qdi/^  when  a  is  regarded  as  con- 
stant, on  the  supposition  of  the  constancy  of  dx  and  dy^  or 
that  X  and  y  are  the  independent  variables,  we  must  regard 
dz  as  also  being  constant;  consequently,  we  shall  have,  in 
this  way,  dpdx-\-dqdy  =  0,  in  which  djj  and  dq  stand  for  the 
total  differentials  of  ^  and  q. 

Since  (see  p.  220) 

dp  =  ~-  dx  -\-  ~  dy  =z  rdx  +  sdy 

and  ^9.^=^  -f  ^^  -^  -^    ^^'J  =  ^dx  +  tdy, 

we  shall  have  the  equation 

dpdx  +  dqdy  =  rdxP  +  2sdxdy  +  tdy^^  =  0. 

Because  dx  and  dy  are  constant,  by  taking  the  total  dif- 
ferentials of  this  equation,  we  shall  have 

d^pdx  +  (f-qdy  —  drdx"  +  Msdxdy  +  dtdy^  =  0. 
If  we    ut 


then,  since 


and 


dr  _ 
dx 

=  u, 

dr  _ 
dy 

ds 
~  dx~ 

% 

ds 
dy~- 

dt 
~d^ 

=  w, 

and 

dt 

dy 

dr^ 

dr 
~  dx 

dx  + 

ds  = 

ds 

""  dx 

dx  + 

ds   , 

dy'y^ 

dt  = 

dt 
""  dx 

dx  + 

dt   ^ 

TWISTED   SURFACES.  227 

tlie  preceding  equation  is  reducible  to 

dydar  +  drqdif-  —  drdx^  +  Idsdxdy  +  dtdy^  « 

=  udx^  +  Zvdx'^dy  +  Zwdxdy^  +  u'd]f'  =  0, 
or,  we  sliall  have 

an  equation  of  the  third   order  of  partial  differential  co- 
efficients of  twisted  surfaces. 

From  dx  =  adz  and  dy  ^=^(1)  (a)  adz,  by  division,  we  get 

-—-  =  — -^ ,  which  must  clearly  be  the  same  as  found  from 

dx  a  -^ 

or  Its  equivalent      y  +  _  _  =  _ -, 

whose  solution  gives 

dy  _  -  s±  i/{&^-rt)  _    ,^ 
dx-  t  -  "^  ' 

consequently,  we  shall  have  —^  =  a' ^   or   0  {a)  =  aa',  and 

the  remaining  eqaation  becomes,  by  substitution  (see  Monge, 

p.  198), 

u'a'^  +  Zim""  +  ^va'  -\-  u  ^  ^. 

If  the  three  directrices  are  so  given  as  to  enable  us  to  find 
the  forms  of  V  («),  0  («)'  ^^^  ^  W?  *-^^^  ^7  finding  the  value 
of  a  in  one  of  the  equations 

a?  =  as  +  V'  {d)     and     y=-z4*  {(')  +  ^  («)? 
and  substituting  it  for  «,  in  the  other,  as  at  p.  225,  we  shall 
have  an  equation  in  x,  y,  and  2,  for  the  equation  of  the 
twisted  surface,  and  what  is  called  the  integral  of  the  equa- 


228  TWISTED  SURFACES. 

tion  of  partial  differential  coefficients  of  the  third  degree, 
given  on  p.  227 ;  if,  however,  a  can  not  be  eliminated  from 
the  equations,  the  equations,  in  their  undetermined  form, 
must  be  taken  for  the  integrals. 

KEMARK& — It  is  manifest  that  whatever  may  be  the 
natures  of  the  directing  lines,  we  may  proceed  in  much  the 
same  way,  as  has  been  done,  to  find  the  equation  of  the 
twisted  surface  described  by  the  motion  of  the  generatrix. 


SECTION  IX. 

CURVATUEE  OF  SURFACES,  AND  CURVES  OF  DOUBLE 
CURVATURE. 

(1.)   Curvature  of  Surfaces. — ^Let 

z'  —  z  =:p  ix'  —  x)-{-q{f  —  y) 

represent  the  equation  of  a  tangent  plane  at  a  point  of  a 
curve  surface  whose  co-ordinates  are  a?,  y,  and  z]  then,  from 
what  is  shown  at  p.  207, 

»-X+^(2-Z)  =  0    and    y -Y  +  q{z -Z)  =  0, 

are  the  equations  of  the  normal  to  the  curve  surface,  at  the 
same  point.  By  taking  the  differential  of  the  tangent  plane, 
supposing  a?,  y,  2,  alone  to  varj,  we  have  dz  ~jpdx  +  qdy^ 
and  from 

^^=^i^^  +  |^2/    and    dq  =  ^£dx^r^f^dy, 

or  (see  pp.  220  and  226), 

dp  =:  rdx  +  sdy     and     dq  =  sdx  +  idy. 

Taking  the  differentials  of  the  normals,  supposing  X,  Y,  Z, 
not  to  vary,  we  have 

dx  +  fdz  -]-  {z  —  Z)  dp  — 

dx  4-  p^dx  +  pqdy  -\-  (z  —  Z)  {rdx  +  sdy)  =  0, 

and  dy  +  pqdx  +  q^dy  -h  {z  —  Z)  (sdx  +  tdy)  =  0 ; 

which  are  equivalent  to 


280  CURVATURE    OF  SURFACES. 

l+y+(3-Z)r+to  +  (3-Z).]g  =  0, 

and      jpq  +  {z-Z)8  +  [1  +  f  +  {2-  Z)f]'^^=0, 

dij 
Eliminating  -j-  fi'om  these,  we  have 

[1  +y  +  (2  -  Z)  r]  [1  +  r  +  (2  -  Z)  i!] 

or  (2  -  ZY{rt  -  s")  +  (2  -  Z)  X 

[(1  +  q')r-2j>qs  +  (1  +y)  j5]  +y  +  ^2  ^  1  =  0; 

and  the  elimination  of  s  —  Z  from  the  same  equation,  gives 


©'[(i.^V-i>.O.J 


[(1  +  q')r-  (1  +i>^)  ^]  -  (jy^  +  1)  s  +pqr  =  0. 
These  formulas  may  be  much  simplified  by  supposing  the 
tangent  plane  at  the  point  (a?,  y,  z)  to  be  parallel  or  coinci- 
dent with  the  plane  a?,  y,  imagined,  to  assist  the  imagination, 
to  be  horizontal,  the  concavity  (or  hollow)  of  the  surface 
being  turned  upward,  and  the  axis  of  z  vertical,  its  positive 

dz  dz 

value  being  reckoned  upward ;  then,  ^  =  —  and  q  =  -,—  will 

evidently  be  reduced  to  naught,  and  the  formulas  will  be 
reducible  to 

and  (fy  +  !:^^^_l=0. 

\ax/  s     ax 

Solving  these  equations  by  quadratics,  we  shall  have 


^  2  irt  -  s') 


CURVATURE  OF  SURFACES.  231 

and 


dy  _—  jr-t)  ±  V{r  —  tf  +"4.y' 


dx  2s 


which,  on  account  of  the  ambiguous  signs,  clearly  show  that 
s  —  Z  and  -^ ,  each  admit  of  two  values. 


Because       dy  ^-^.-t)  +  ^/(r-tf +  if 
dx  2s 


and  ^'  =  -(r-t)-V{r-tf  +  i^  _ 

dx  2s 

evidently  represent  the  natural  tangents  of  the  angles  which 
two  vertical  sections  of  the  surface,  by  planes  through  the 
axis  of  2,  make  with  the  plane  of  the  axes  of  x  and  s; 

by  taking  the  product  we  shall  have  ~  x  -^  =  —  1,  conse- 
quently, if  A  stands  for  the  angle  whose  tangent  is  -^, 

A  +  90°  must  stand  for  the  angle  whose  tangent  is  -j- , 

ctx 

A       J.      /  A    .  AAox      sin  A         cos  A  ^ 

smce  tan  A  x  tan  (A  +  90  )  = r  x  -. — r  =  —  1, 

^  ^      cos  A       —  sm  A 

and,  of  course,  the  two  planes  passing  through  the  axis  of  z 
cut  each  other  perpendicularly. 

If   we  represent  z  —  Z   by   E,   we   shall  have  for  the 


equation      z-Z  =  -^ 2{rt-s^)     

the  transformed  equation 


2  (rt  -  s') 

which  clearly  represent  the  radii  of  curvature  of  the  pre- 
ceding perpendicular  sections  at  their  point  of  contact  with 


232  CURVATURE    OF  SURFACES. 

the  plane  of  the  axes  of  x  and  y.  Representing  these  radii 
separately  by  R  and  R',  and  taking  the  sum  of  their  re- 
ciprocals, we  have 

E      R'      r  +  t-V{r-(f  +  4:^ 


r  +  t-\-  V{r  —  tf  +  4v 

_,  dp       d^z         1     ^       do       d}z 

Because  t*  =  -f-  =  -7-^    and    ^  =  -r^  =  — ^ , 
dx       ddcr  ay       ay 

dx"  dij 

and  that  -^3-  and  ~-  are  clearly  the  radii  of  curvature  of 

the  vertical  sections  of  the  surface  which  pass  through  the 
axes  of  X  and  y^  it  ^clearly  follows  that  7'  -\- 1  expresses  the 
sum  of  the  reciprocals  of  these  radii.  Consequently,  since 
the  position  of  the  axes  of  x  and  y  in  the  plane  of  x,  y,  is 
arbitrary,  it  clearly  follows  that  the  sum  of  the  reciprocals 
of  the  radii  of  curvature  of  any  two  vertical  planes  through 
the  axis  of  s,  which  cut  each  other  perpendicularly,  is  always 

equal  to  -p  +  ^7  ;  and  of  course  the  sum  of  the  reciprocals 

of  the  radii  of  any  two  such  sections,  is  always  equal  to  the 
sum  of  the  reciprocals  of  the  radii  of  any  other  two. 

If,  according  to  custom,  we  represent  the  curvature  of  the 
circumference  of  a  circle  by  the  reciprocal  of  its  radius,  we 
shall  have  the  sum  of  the  curvatures  in  any  tvjo  vertical 
sections  that  pass  through  the  axis  of  2,  and  cut  each  other 
perpendicularly^  equal  to  the  sum  of  the  curvatures  iii  any 
other  two  vertical  sections  that  pass  through  the  axis  of  s, 
and  cut  each  other  perpendiculariy.  It  is  hence  clear,  that 
if  the  curvature  in  one  of  two  perpendicular  planes  is  a 
maximum,  that  it  must  be  a  minimum  in  the  other  plane. 


CURVATURE  OF  SURFACES.  233 

and  vice  versa.  It  is  also  clear  tliat  E  and  E' — called  the 
principal  radii — are  such,  that  the  first  is  less  than  any 
other  radius  of  curvature,  while  the  other  is  greater  than  any 
other  radius. 

If  we  take  the  principal  sections  for  the  axes  of  co-ordi- 
nates, then 
drj        -{r-t)+  V{r-  tf  +  ^s^  _   25 


dx  2s  r-t-\-  V{r-tY  +  4.s' 

and 


dy'  _   -{r-t)-  V{r~tY  +  4:S^ 2s_ 


dx  2s  r—t—  V{r  —  if  +  4/- 

given  at  p.  230,  may  be  taken  for  the  tangents  of  the  angles 
which  a  pair  of  perpendicular  planes  through  the  axis  of  z 
makes  with  the  first  of  the  principal  planes  through  the  axis 
of  X. 

If  the  perpendicular  planes  are  brought  to  coincidence 

with  the  axes  of  co-ordinates,  we  shall  have  ~-  :=  0 ;  and  of 

course,  from  the  first  of  the  preceding  formulas,  we  must 
have  5  =  0.     Hence,  putting  s  =  0  in 


^^rJ^-Vp^P^    and  r,  ^r+^+  i/(.-.y+4,^  _ 
2  {rt  —  s^)  2  {rt  —  6'-) 

they  will  become  E  =  -    and    E^  =  -  for  the  radii  of  cur- 
-^  r  t 

vature  of  the  principal  perpendicular  sections. 

Supposing  E^'  to  stand  for  the  radius  of  a  perpendicular 

section  through  the  axis  of  ^,  which  makes  an  angle  whose 

tangent  =  ~  with  the  axis  of  a?,  then  we  shall  clearly  have 

_,„        dm?  +  dip'        dx^  +  dip- 


d?z  rdx^  H-  tdy^  ' 


284  CURVATURE  OF  SURFACES. 

since  5  =  0,   or  K  '  =  - 


r  +  t 


\dx) 


wliich,  bj  supposing  -^  =  tan  0,  becomes 
^„^   l+tan»(A   _  1 


r  -\-t  tan-  0       7"  cos^  </>  +  ^  sm-  «/> 

whicli,  by  putting  for  r  and  ^  their  values  -^  and  :p7 ,  is 
easily  reduced  to 

T>// 5^5 

K'  cos^  0  +  K  sin^  <i«) ' 

,  .  ,     .  1         cos*  (p       sin*  <^ 

which  gives        ^7-,  =  -^-  +  -^—  ; 

which  clearly  show  that  if  K  =  R',  we  shall  have  W  =  E, 
so  that  the  radii  of  all  the  sections  through  the  axis  z  equal 
each  other. 

If  R  is  positive,  and  R'  negative,  the  preceding  value  of 
"R"  will  be  reduced  to 

R'R  R'R 


R 

and 


—  R'  cos^  <A  +  R  sin-  <i>  ~  W  cos"-^  </>  —  R  sin^  <!> 
1    _  cos*  0       sin*  <f> 
W  ~  ~R  ~W  ' 


noticing,  that  what  is  here  done  corresponds  to  a  circular 
wheel  with  a  groove  in  its  circumference,  R'  representing  the 
radius  of  the  wheel  whose  convexity  is  turned  upward,  and 
R  the  radius  of  the  groove  whose  convexity  is  turned 
downward,  and  its  concavity  upward. 

If  R  sin*  <^  =  R'  cos^  <p    or    tan  0  =  ±  V  W  ' 

•p/T> 

we  shall  have  'R"  =  -jr-  —  infinity ; 


CURVATURE    OF  SURFACES.  235 

consequently,  tan  0  =  y  =5-     and     tan  0  =r  —  y  — - 

indicate  two  riglit  lines  on  tlie  surface  drawn  to  make  angles 
with  the  axis  of  x,  or  width  of  the  groove,  passing  through 
its  center  and  making  angles  with  it,   whose  tangents  are 

|/  -5-  and  —  y  tp-  on  the  positive  and  negative  sides  of  x 

positive,  and  on  the  positive  and  negative  sides  of  a?  negative; 
the  surface  between  these  tangents  being  clearly  concave, 
while  the  remaining  part  of  it  is  evidently  convex,  so  that 
the  tangents  separate  the  concavity  and  convexity  of  the 
surface  from  each  other. 

Supposing  the  right  line  x'  is  drawn  from  the  origin  of 
the  co-ordinates  in  the  plane  of  x,  y,  to  the  surface,  so  as  to 
make  the  angle  <f)  with  the  axis  of  x  (see  Young,  p.  183) ; 

,         ,  .  /cos-  0      sin^  <t>\    ,._j 

then,  by  assummg  z  =  [^^-  ±  -2^)  ^  , 

since  cos^  (px'^^  =  x^    and     sin^  (jix"^  =  y^,  ' 

we  shall  have  ^  ^  Ie  ^  ^ ' 

which  is  the  equation  of  the  surface  of  a  paraboloid  of  the 

second  order. 

T^  /cos"(/)   .   sin^</)\     ,, 

From  .^(_^^.±_),,^^ 

I.  ^"  2RR' 

we  have        —  —  ^, ^-r  t-w-^-t-t  =  2R'', 

z       R  cos^  </>  ±  R  sm^  <f>  ' 

and  R''  is  the  radius  of  curvature  of  a  vertical  section  of  the 
paraboloid,  at  the  point  whose  co-ordinates  are  x  and  y^  as  it 
clearly  ought  to  be. 

Hence  we  perceive  how  to  measure  the  curvatures  at  any 
proposed  point  of  a  surface  by  those  of  the  paraboloid,  and 


236  THEOREM    OF  MEUSNIER. 

that  whether  the  principal  curvatures  have  the  same  or  con- 
trary directions. 

Eemarks. — 1.  Resuming  R'^  =    '      ,^   ^  ,  from  page  233, 

di^ 

and  representing  VdJ^  +  dy"'  by  ds^  it  will  become  R''  —  -,^- ; 

which  is  the  radius  of  curvature  in  a  normal  section  to  the 

^  curve  surface  at  its  point  of  contact  with  the  plane  of  the 

axes  X  and  y.     Suppose  now  a  j^lane  making  the  angle  0 

with  the  normal  section,  also  touches  dn  at  the  origin  of  the 

co-ordinates ;  then,  it  is  clear  that  -z^,  will  be  the  radius  of 

Cu  Z 

curvature  in  the  oblique  section  with  the  curve  surface,  in 
which  d^z'  corresponds  to  d^z  taken  in  the  normal  plane. 
Because  d?z'  and  d^z  are  clearly  the  hypotenuse  and  side  of  a 
right  triangle  having  0  for  their  included  angle,  we  shall 

d'z 
have  d?z'  cos  Qz^d^z    or    d/z'  — 


cos  6/' 


which  reduces  -^-7    to     -rp  x  cos  0  =  R'^  x  cos  6  ; 
drz  d-z  ' 

consequently,  the  radius  of  curvature  in  the  oblique  section 
equals  the  (orthographic)  projection  of  R^'  on  the  plane  of 
the  oblique  section,  which  is  called  the  Theorem  of 
Mcusnier. 

2.  Resuming  the  equations  of  the  normal  from  p.  229,  and 
putting,  with  Monge, 

g=rt-^,  h=={l  +  q')r-2pq8  +  {l+f)t,  and  ¥=^f+q'^-l 

in  the  fourth  equation  at  p.  230,  we  shall  have 

aj-X+^(s-Z)  =  0,    7/-Y  +  ^(5-Z)  =  0, 


THEOREM    OF    MEUSNIER.  237 

and  (^-Z)^  +  -(3-Z)  +  -  =0, 

whose  solution  ogives  s  —  Z  =  -, — ■ — -—7^ ir^nk  • 

^  ±  \\'('''  —  ^gKT) 

Supposing  E  to  be  the  radius  of  a  sphere  that  touches  the 

curve  surface  at  the  point  (a?,  y,  2),  having  X,  Y,  Z,  for  the 

co-ordinates  of  its  center,  we  shall  have 

for  the  equation  of  the  spheric  surface,  which,  by  substitu- 
tion from  the  equation  of  the  normal,  becomes 

E;.:.  {f  +  ^2  +  1)  (3  -  ZJ  ^¥{z-  Z/; 
consequently,  from  the  substitution  of  the  preceding  value 
of  s  —  Z  in  this,  we  readily  get 

p  _  2^^ 

A  ±  |/(^'  -  ^g^")     ■ 
for  the  two  radii  of  curvature  at  any  proposed  point  of  the 
curve  surface. 

To  illustrate  what  has  been  done,  we  will  apply  it  to  find 
the  radii  of  curvature  of  the   surface,  whose   equation  is 


>  =  f- 

Here  we  have  -f^ 

dx 

=^  =  |-     and 

dz  _ 
dy~ 

■.q  = 

X 

^  A' 

which  give 

B 

r^   +    f   +    A^ 

~          A^ 

and  from 

d^z  __dp  _     _ 

d^  ~~  dx~     ~ 

0, 

since  p  is  not  a 

function  of  x^  and,  in 

the  same 

way, 

dq_ 
dy 

2^  =  0,     but     5  =  ^: 
dy 

dx 

1 
~A 

? 

238  THEOREM    OF  MEUSNIER. 

wbicli  give  h=  —  2pqs  = ^ , 

and  they  also  reduce  g  =  rt  —  ^  to  —  5^  =  —  -— . 

Hence  the  equation    (2  —  Z)  s  +  -  (2  —  Z)  H —  =  0, 

is  easily  reduced  to 

(2_Z/  +  ^^(2-Z)  =  ^  +  y'  +  A', 

■whose  solution  gives 

„       -mi±  4/[A*  +  A'  (a;'  +  y')  +  !^f\ 
z-Z= J , 


or         Zi  —  s  = ^^ 

which  gives 


or  R  = 


■       (A^  +  a^  4-  2/^)^ 


for  the  expression  of  the  radii  of  curvature,  at  any  proposed 
point  of  the  given  surface. 

If  in  the  preceding  value  of  R  we  put  a;  =  0  and  y  =  0, 
we  get  R  =  ±  A  or  R  =  A  and  R  =  —  A  for  the  radii  of 
curvature  at  the  origin  of  the  co-ordinates,  which  is  clearly 
that  of  the  vertex  of  the  given  surface ;  since  these  radii 
have  contrary  signs,  it  is  manifest  that  the  principal  curva- 
tures of  the  surface  at  its  vertex,  are  turned  in  opposite 
directions,  and  it  is  manifest  that  like  conclusions  are  ap- 
plicable to  any  other  point  of  the  proposed  surface,  but  their 
magnitudes  are  not  equal,  as  at  the  vertex. 


CURVES  OF  DOUBLE  CURVATURE.         239 

It  is  easy  to  perceive,  that  by  making  analogous  substitu- 
tions to  those  made  in  the  equations  for  the  radii  vectores  in 

the  quadratic  equation  in  -~  given  at  p.  230,  we  may,  after 

the  manner  of  Monge,  at  pp.  121,  &c.,  of  his  "Application 
de  r Analyse  a  la  Geometric,"  proceed  to  find  the  integral 
of  the  equation,  and  thence  to  trace  out  the  lines  of  curva- 
ture on  any  proposed  surface,  together  with  the  correspond- 
ing radii  of  curvature.  We  shall  not,  however,  attempt  to 
do  this,  but  shall  satisfy  ourselves  with  the  following  obser-, 
vations. 

Thus,  from  what  is  shown  at  p.  230,  it  results  that  there 
are,  at  any  point  of  a  curve  surface,  tw©  lines  of  curvature 
at  right  angles  to  each  other,  such,  that  the  successive  nor- 
mals to  the  surface  in  each  intersect  each  other  and  form  a 
developable  surface;  the  line  in  which  the  successive  normals 
intersect  being  called  the  edge  of  regression  of  the  develop- 
able surface^  while  the  lines  in  which  the  developable  sur- 
faces cut  the  proposed  surface  are  called  lines  of  curvature, 

(2.)  Curves  of  Double  Curvature. — In  treating  of  curves 
of  double  curvature,  it  will  be  sufficient  to  regard  them  as 
consisting  of  indefinitely  small  arcs,  regarded  as  straight 
lines ;  such  that  (in  general)  no  more  than  two  successive 
arcs  can  lie  in  the  same  plane. 

Suppose  then  x  —  x'-^K  (y  —  y^)  +  B  {z  —  z')  =  0,  to 
represent  the  plane  of  any  two  successive  sides  of  the  curve, 
having  x,  y,  z,  for  the  rectangular  co-ordinates  of  the  first 
extremity  of  the  first  of  the  two  successive  sides,  and  x\  y\ 
z\  for  the  co-ordinates  of  any  other  point  of  the  plane ;  then 
dx^  dy^  and  dz^  being  the  differentials  of  x^  y,  and  ^,  we  shall 
have  dx  +  Ady  +  ^dz  =  0,  when  we  pass  from  the  co-ordi- 
nates of  the  first  extremity  of  the  first  (short)  side  to  those 


210        CURVES  QF  DOUBLE  CUBVATUBE. 

of  its  second  extremity,  or  those  of  the  first  extremity  of 
tlie  second  side ;  and  in  passing  from  the  first  to  the  second 
extremity  of  the  second  (short)  side,  we  in  like  manner  get 

cT-x  +  AcPy  -f-  BcT'z  =  0. 
From  the  last  two  of  these  equations  we  get 

.  _  dzcPx  —  dxd-2       ,   TK  —  ^^^^'y  ~  ^y^^ . 

~  dyd^z  —  dzd^y '  "  dijd-z  —  dz^-y ' 

and  from  the  substitution  of  these  values  of  A  and  B  for 
them  in  the  equation  of  the  plane,  it  is  readily  reduced  to 
the  form 

{x  —  a?')  {dy^z  —  dzdry)  +  {y—y')  {dzd/x  —  dxdrz)  + 
{z  —  z')  {dxd-y  —  dyd^x)  =  0, 

which  is  sometimes  called  the  osculating  plane  of  the  point 

(iP,  y,  4 

Supposing  R  to  he  the  radius  of  a  circle  passing  through 
the  extremities  of  the  same  two  successive  short  sides,  and 
that  the  point  (a?',  y\  z')  is  taken  at  the  center,  we  shall, 
from  the  nature  of  the  circle,  have  the  equation 

W  =  {x  -  xj  4-  Cy  -  yj  +  (^  -  ^y 

for  the  first  extremity  of  the  first  short  side;  which,  in 
passing  to  the  second  extremity  of  the  first  side,  gives 

{x  —  x')dx  +  {y  —  y')  dy  +  {z  —  z')  dz  =  0, 

and  this,  when  we  pass  to  the  second  extremity  of  the 
second  side,  gives 

{x  -  x')  drx  +  {y-  y')  dhj-^{z  -  z')  d'z  +  dx"  +  dif  +  dz'. 

If  ds  represents  the  length  of  the  first  side,  since  dx^  dy^  dz^ 
arc  clearly  the  projections  of  fZ.9  on  the  axes  of  a?,  y,  and  ^,  it 
is  easy  to  show  that  we  must  have 

dsr  +  dy-  +  dz^  —  ds^ ; 


CURVES  OF  DOUBLE  CURVATURE.        241 

consequently,  the  last  of  the  preceding  equations  becomes 

(x  -  x')  d'x  +  (y  -  2/0  ^V  +  (^  -  ^0  ^'2  +  ^«'  =  0. 
Bj  successively  eliminating  z  —  z'^y  —  y\  and  a;  —  a?',  from 
the  preceding  equation,  by  means  of  the  equation 

{x  —  x')  dx  +  {y  —  y')  dy  ^{z  —  z')  dz  =  0, 

we  have  the  equations 

{x  —  x')  {dx(^z  —  dzd'^x)  -{-  {y  —  y')  {dydh  —  dzd^y)  =  dzds% 
{x  —  a?')  (dxd}y  —  dyd}x)  +  (z  —  z')  {dzd^y  —  dyd^z)  =  dyds% 

and 

{y  —  y')  {dyd^x  —  dxd^y)  -\-  {z  —  z')  {dzd^x  —  dxd^z)  =  dxds\ 

Hence,  supposing  x\  j/,  z',  in  the  osculating  plane,  to  corre- 
spond to  the  center  of  the  circle,  by  adding  its  square  to  the 
squares  of  the  three  preceding  equations,  because  the  double 
products  destroy  each  other,  we  shall  have,  since 

{x-xy  +  (2/  -  yj  +  {b-z'Y  =  R' 

and  dz^ds^  -f  dy^ds''  +  dx'ds^  =  ds\ 

■D2  _. 

{dxd^y  —  dyd^xf  +  {dxdC-z  —  dzd'xf  +  {dyd^z  —  dzd?yY  * 

for  what  is  sometimes  called  the  square  of  the  radius  of  the 
absolute  curvature,  corresponding  to  the  point  (a?,  y,  z). 

From  the  development  of  the  squares  in  the  denominator, 
and  omitting  the  factor  dx^  +  dy^  +  dz^  =  ds^^  that  is  common 
to  the  numerator  and  denominator  of  the  resulting  fraction, 

we  have  R°  =   ,.->  >,  ,   ^02.^0 -irz. 

d~x-  +  d-y^  -f  a-z-  —  d's^ 

If  ds  equals  its  successive  side,^  ds  is  constant,  and  d^s  =  0] 
consequently,  we  shall  have 

~  d'i^  +  dy -{- (fz"^' 
11 


242        CURVES  OF  DOUBLE  CURVATURE. 

whicli  can  readily  be  deduced  from  the  simplest  principles 
of  geometry,  and  that,  whether  the  proposed  curve  is  of 
single  or  double  curvature. 


Thus,  lei  ah  ^zhc  —  ds  represent  two  successive  sides  of 
the  polygonal  curve,  having  the  circular  arc  whose  center  is 
<?,  passing  through  their  extremities ;  then,  oa  =  oh  —  OG=zB, 
will  clearly  be  the  radius  of  curvature  of  the  proposed 
curve  corresponding  to  the  point  a,  which  we  shall  suppose 
to  have  a?,  y,  and  s,  for  its  rectangular  co-ordinates.  It  is  clear 
that  the  equal  straight  hues  ah  and  ho  subtend  the  equal 
arcs  ah  and  hc^  which,  when  ah  and  ha  are  indefinitely  small, 
will  differ  insensibly  from  them.  If  ah  is  produced  to  d^  so 
as  to  make  hd  =  ah,  then,  drawing  the  right  line  cd,  it  will 
evidently  be  parallel  to  oh. 

It  is  also  manifest  that  the  triafigles  chl  and  ohc  or  oha  are 

equiangular,  and  give  the  proportion 

h(^      d.r 
do  :  ch  ::  ch  :  ho ;    which  ffives     R  =  —^  =  —-. 
°  cd       cd 

Also,  dx,  dy^  and  dz,  the  differentials  of  a?,  y,  and  z,  the  co- 
ordinates of  the  point  a,  are  -evidently  equal  to  the  projec- 
tions of  ah  or  ds  on  the  axes  of  a?,  y,  and  2,  which  are  clearly 
equal  to  the  projections  of  hd  on  the  same  axes;  and,  in  like 
manner,  the  differentials  of  the  co-ordinates  of  the  point  h 


RADIUS  OF  SPHERICAL  CURVATURE.        243 

equal  the  projections  of  hc^  or  the  (algebraic)  sum  of  the 
projections  of  hd  and  do,  on  the  axes  of  x^  y,  and  z. 

Hence  (see  p.  2),  since  the  differentials  of  the  co-ordinates 
of  the  point  h  diminished  by  those  of  the  point  «,  equal 
the  second  differentials  of  the  point  a,  we  get  d^x.,  d}y^  and 
o?-3,  equal  to  the  sum  of  the  projections  of  hd  and  do  dimin- 
ished by  those  of  aj,  on  the  axes  of  x^  y^  and  z ;  conse- 
quently, since  the  projections  of  hd  are  destroyed  by  those 
of  a  J,  it  clearly  results  that  d^^ic^  d}y^  and  d^z^  are  the  projec- 
tions of  cd  on  the  axes  of  a?,  y,  and  z. 

Hence,  from  the  nature  of  the  projection,  it  being  the 
orthographic  (or  orthogonal)  projection,  we  shall  have 
cd^  =  d'xr  +  dhf  +  d-z'' ; 

d.^  ds* 

which  reduces    E^  =  — ^    to     E^  = 


ccP  cZV  +  dy  +  d^z^' 

which  agrees  with  what  is  shown  at  p.  241. 

Thus  far  x,  y,  and  z,  have  been  regarded  as  being  inde- 
pendent of  each  other ;  we  now  propose  to  consider  y  and  z 
as  being  functions  of  a?,  expressed  hj  y  —  ip  {x)  and  s  ==  <^  {x\ 
the  projections  of  the  curve  of  double  curvature  on  the 
planes  of  a?,  ?/,  and  a?,  2,  and  shall  assume 

7^  ^{x-  xj  -f  (y  -  yj  +  {z-  zj 

for  the  radius  of  a  spherical  surface,  called  the  radius  of 
sj)herical  curvature^  supposed  to  pass  through  any  four  suc- 
cessive angles  of  the  polygonal  line. 

If  a?,  2/,  s,  represent  the  (rectangular)  co-ordinates  of  the 
first  of  the  successive  angles,  by  passing  to  the  second  angle 
we  get  the  differential  equation 

{x  —  x')  dx  +  {y  —  y')  dy  -\-  {z  —  z')  dz  =  0, 


2rl4  CURVES   OF  DOUBLE   CURVATURE. 

which  is  clearly  the  equation  of  a  plane  at  right  angles  to  the 
first  indefinitely  short  side  that  connects  the  first  two  of  the 
successive  angles,  and  passes  through  the  center  of  the 
spherical  surface ;  or,  putting 

dy  _  dij)  (x)         ,     dz  _d((>  (x) 
dx  ~     dx  dx~    dx    '' 

equal  to  P  and  Q,  the  equation  will  be  more  simply  expressed 

by  x-x'-\-{y-y')V  +  {2-z')Q,  =  0. 

Representing  -^  and  ~j-  by  P'  and  Q',  and  taking  the  dif- 
ferential of  this  equation,  we  have 

(y  -  2/0  P'  +  (^  -  ^')  Q^  +  F  +  Q^  + 1  =  0, 

and  from  this,  by  putting  -y-  and  -y-  equal  to  V  and  Q'', 

we  in  like  manner  get 

(y  -  yO  V"  +{z-  z')  Q''  +  3  (PP'  +  QQO  =  0. 

The  first  two  of  these  equations,  since  they  represent 
planes  perpendicular  to  the  first  and  second  short  sides,  and 
that  they  intersect  in  a  right  line,  clearly  show  the  character- 
istics to  be  placed  on  a  developable  surface,  or  to  have  a  de- 
velopable surface  for  their  envelope.  And  it  is  evident  that 
the  three  equations  together  represent  the  edge  of  regres- 
sion, or  the  line  in  which  the  successive  straight  lines  (that 
characterize  the  developable  surface)  intersect,  which  is  evi- 
dently the  line  in  which  the  centers  of  spherical  curvature  lie. 

It  may  be  noticed  that  if  the  origin  of  the  co-ordinates  is 
taken  at  a  point  in  the  curve  having  the  axis  of  x  for  a  tan- 
gent at  the  origin,  the  preceding  formulas  will  be  much 
simplified.  For,  at  the  origin  we  shall  have  a?,  y,  2,  each  equal 
to  naught,  and  P  and  Q  are  also  reduced  to  naught,  or  be- 
come infinitesimals.     Hence,  we  shall  have 


CURVES  OF  DOUBLE  CURVATURE.         245 

x'  =  0,     2/T'  +  b'Q'  =  1,     and    yT^'  +  zW  =  0 ; 
consequently,  we  sliall  have 

P'^2  _|_  Q'/2 

and  thence  W  =  y"'  +  2''  reduces  to  R-  =  /p  rj^^  _  p//n/\2  • 

Again,  if  in  the  first  two  of  the  formulas  we  put  -0  {x)  and 
9  (a?)  for  y  and  s,  and  eliminate  x  from  the  results,  we  shall 
obtain  an  equation  in  x\  y\  and  z\  for  the  equation  of  the 
developable  surface  noticed  above. 

li  3  —  z'  is  eliminated  from  the  same  three  equations,  and 
^  {x)  is  put  for  y  in  the  two  resulting  equations,  then  the 
elimination  of  x  from  these  equations  gives  an  equation  in 
terms  of  y'  and  x'  for  the  equation  of  the  projection  of  the 
edge  of  regression  on  the  plane  of  the  axes  of  x  and  y. 

Similarly,  by  first  eliminating  y  —  y'  from  the  equations, 
putting  (f>  {x)  for  2  in  the  results,  and  eliminating  a?,  we  shall 
get  an  equation  in  x'  and  2'  for  the  equation  of  the  projec- 
tion of  the  edge  of  regression  on  the  plane  of  the  axes  of 
X  and  2. 

If  from  any  point  in  the  common  section  of  the  first 
two  perpendicular  planes  a  straight  line  is  drawn  in  the 
second  plane  to  the  second  short  side  of  the  curve  of 
double  curvature,  and  produced  in  the  opposite  direction 
to  meet  the  line  of  common  section  of  the  second  and 
third  planes,  and  a  line  drawn  from  the  point  thus  found 
to  the  third  short  side  of  the  curve,  and  produced  in  the 
opposite  direction  to  meet,  as  before,  the  line  of  common 
section  of  the  third  and  fourth  planes,  and  so  on  ;  then,  a 
straight  line  drawn  from  the  first  (assumed)  point  to  the  first 
short  side,  and  continued  as  a  curve  in  the  opposite  direction 


246         CURVES  OF  DOUBLE  CURVATURE. 

through  the  points  found,  will  clearly  represent  the  evolute 
of  the  proposed  curve,  regarded  as  its  involute.  (See  Sec. 
VI.  p.  170.)  It  is  hence  easy  to  perceive  that  the  proposed 
curve  of  double  curvature  has  an  unlimited  number  of  evo- 
lutes. 

(j[y^  y  y^ 

It  is  manifest  that  -j-y  = —,  represents  a  tangent  of  an 

evolute  of  the  proposed  curve,  when  projected  on  the  plane 
of  ar,  y,  and  that  by  eliminating  z  —  z'  from  the  equations 

x-x'  +  (y -2/')  P  -f  (5  _  s^)  Q=:  0 

and  {y-y')  P'  +  (2  -^0  Q  +  P  +  Q^  +  1  -  0, ' 

and  putting  i/>  (a?)  for  y  in  the  resulting  equation,  and  in 

«¥  ^  y-y' 

dx'       X  —  x'  ^ 

then,  eliminating  x  from  these  equations,  we  shall  obtain  a 
differential  equation  in  x'  and  ?/',  whose  integral,  according 
to  the  principles  of  the  Integral  Calculus,  will  involve  one 
arbitrary  constant  The  constant  will  enable  us  to  make  the 
evolute  pass  through  any  proposed  point ;  for  if  the  co- 
ordinates of  any  point  a?',  y\  and  z\  are  represented  by  a^  h, 
&c.,  by  putting  a  and  7j  for  a?'  and  y'  in  the  integral,  we 
easily  get  the  value  of  the  constant,  and  thence  the  integral 
is  determined,  so  that  the  projection  of  the  evolute  on  the 
plane  x,  y,  passes  through  that  of  the  proposed  point ;  and  in 
a  similar  way  the  projection  of  the  same  evolute  on  the 
plane  a?,  s,  may  be  determined,  and  the  evolute  will  be  found 
as  required. 

For  an  example  we  will  put 

y  =  x    and    z  =  x^    for    y  =  V'  (^)     and    s  =  0  (a?), 

which  give     ^  =  P  =  1,      ^=Q=.2x,      ^  =  0  =  F\ 
^         dx  dx  ^      dx  ' 


CURVES  OF  DOUBLE  CURVATURE.         247 

^  =  2=:.Q',     F^  =  0,     and    Q'^  =  0. 

Hence  tlie  formulas  given  at  p.  244  become 

x  —  x'-\-y  —  y'-{-^x{z  —  z')  =  0, 

2  —  s' +  2i»2  +  1  =  0, 

and  Q  =  2a?=:0, 

of  whicli  tlie  first  two  give  tlie  developable  surface,  and  all 
tbree  give  the  edge  of  regression.  If  tlie  values  of  y  and 
z  are  put  for  them,  in  the  first  two  of  these  equations,  they 
will  become 

2x  —  x'  —  y'  +  2x^  —  2xz'  =  0     and    1  —  z'  -\- Sx""  =  0 ; 

consequently,  putting  a?  =  0  in  these  equations,  we  have 
a?'  -j-  y^  —  0  and  z^  =  1  for  the  equations  of  the  edge  of 
regression. 

Eliminating  x  from  the  first  two  of  the  preceding  equa- 

"^^—^ — I    =  I — x^^  /    ^^^  ^^^  equation 

of  the  developable  surface,  which  clearly  shows  that  z^  must 
be  positive,  and  not  less  than  1. 

we  put  X  for  y,  we  shall  have 

dy'  {x  —  x'')  =  dx'  {x  —  y'\ 
for  the  equation  of  the  projection  of  a  tangent  to  an  evolute 
on  the  plane  x^  y.     And  eliminating  z'  from  the  equations 

"Ix  —  x'-y'^  2a?^  —  2xz'  =  0     and     1  —  z'  -{- Zar  =^  0, 


we  get 


=  -f^^f; 


consequently,  putting  this  for  x  in  the  preceding  differential 
equation,  it  becomes 


248  POIITTS  OF  INFLECTION. 

for  the  differential  equation  of  the  projection  of  an  evolute 
on  the  plane  a?,  y. 

Hence  we  must  find  the  integral  of  this  equation  and 
determine  its  constant,  so  as  to  suit  the  nature  of  the  case. 

It  may  be  added  to  what  has  been  done,  that  if  a  curve  of 
double  curvature  has  single  curvature  at  one  or  more  of  its 
points,  it  is  said  to  have  lost  one  of  its  curvatures,  and  to 
have  a  single  inflection  at  such  points  ;  while,  if  it  loses  both 
of  its  curvatures  (or  becomes  rectilineal)  at  one  or  more  of 
its  points,  it  is  said  to  have  a  douhle  inflection  at  such  points. 

It  is  manifest  from  the  definition,  that  the  points  of  single 
inflection  in  curves  of  double  curvature,  may  be  found  by 
putting  the  expression  for  the  radius  of  spherical  curvature 
equal  to  oo ,  or  by  putting  its  reciprocal  equal  to  0. 

p//2     I      Q//2 

Thus,  by  putting  t^ojty' T>"C\'\^  *^®  expression  for  E^, 

given  at  p.  245,  equal  to  infinity,  or  putting  its  reciprocal 
equal  to  naught,  we  have  P'Q''  —  P"Q'  =  0,  which,  from 
what  is  done  at  p.  244,  is  the  same  as  to  put 

drydPz  —  il^y(Pz  =  0, 
or  to  find  the  points  of  the  curve  which  satisfy  the  equation 
d^2  _  d^y 
d}z  ~  d-y  ' 
If  this  equation  can  not  be  satisfied  at  any  point  of  a  curve 
of  double  curvature,  it  clearly  can  not  have  a  point  of  sin- 
gle inflection ;  while,  if  it  can  be  satisfied  at  one  or  more 
points  of  the  curve,  it  is  manifest  that  the  curve  may  have 
single  inflections  at  such  points. 

To  find  the  points  of  double  inflection,  we  put  the  radius 


POINTS   OF   INFLECTION.  249 

of  absolute  curvature  of  tlie  curve  of  double  curvature, 
cither  equal  to  0  or  oo ;  consequently,  from  tbe  expression 
for  E"^  given  at  p.  243,  we  bave 

c^V  +  ^y  +  ^V  ==  0  or  00. 

Since  tbis  expression  bas  been  obtained  on  tbe  supposition 
of  tbe  constancy  of  ds^  =  dx^  +  dy^  +  dz'^,  we  put  tbe  differ- 
ential of  tbis  equal  to  naugbt,  wbicb  gives 

dxd^x-^dyd'y-^dzd'B^O     or     drx  =  -  ^^^^-±^^  ] 

wbicb,  put  for  cPx  in  tbe  preceding  equation,  gives 

d^f  +  cT-^^  +  (^^y  +  '^''P'J  =  0  or  00 . 

Because  tbis  expression  consists  of  tbe  sum  of  tbree 
squares,  it  is  evident  tbat  we  must  satisfy  it  by  putting  eacb 
of  its  terms  separately,  equal  to  0  or  oo ;  consequently,  tbese 
conditions  will  be  satisfied  by 

dy  =  0  or  d^y  =  co ^  and  d^z  =0  or  d^z=  a:). 
Tbese  conditions  clearly  follow  from  tbe  projections  of  tbe 
curve  on  tbe  planes  a?,  y,  and  a?,  2,  wbicb  will  manifestly  be 
plane  curves  baving  (eacb)  tbe  same  number  of  points  of 
inflection ;  consequently,  from  tbe  rule  given  at  p.  156,  we 
must  bave 

d^y       „         d'^y  dx        .  ,    d'^z      ^        dx        ^ 

--^=Oor-y^=oo  or-^  =  0  and  —  =  0  or  -^  —^, 
ax  dx  ay  dx  d-z 

wbicb  are  clearly  equivalent  to  tbe  preceding  conditions. 
11* 


INTEGRAL   CALCULUS. 


INTEGRAL   CALCULUS, 


SECTION  L 

(1.)  The  Integral  Calculus  is  the  reverse  of  the  Dif- 
ferential Calculus;  the  object  being  to  find  the  function 
called  the  integral^  from  which  any  proposed  differential 
may  be  supposed  to  have  been  derived. 

Thus,  since  2xdx  and  nx''~'^dx  are  the  differentials  of  ix? 
and  a?"  or  (more  generally)  of  x'^-\-o  and  «"  +  <?',  c  and  c'  be- 
ing called  the  arbitrary  constants^  it  follows  that  x^  and  x^^ 
or,  more  generally,  x^  +  c  and  x^  +  g\  are  the  integrals  of 
the  proposed  differentials. 

In  like  manner,  each  of  the  examples  given-  under  the 
rule  at  p.  5,  when  an  arbitrary  constant  c  (for  generality)  is 
added  to  it,  is  the  integral  of  its  differential;  so  that  the 

most  general  integral  of  ox'^dx  is  a?^  -f  c,  and  that  of  -v-  ^~^dx 

is  T  a?"  +  c  (see  examples  1  and  4). 

Hence,  by  reversing  the  rule  at  p.  5,  it  is  clear  that  if  we 
have  a  differential,  such  that  any  power  of  a  variable  ex- 
pression is  multiplied  by  the  differential  of  the  expression 
under  the  index  of  the  power,  which  may  be  multiplied  by 
one  or  more  constants ;  then,  the  integral  may  be  found  by 
the  followinor 


254  INTEGRAL  CALCULUS. 

RULE. 

Increase  the  index  of  the  variable  expression  by  unity, 
and  divide  by  the  increased  index  and  by  the  differential  of 
expression  under  the  index,  and  add  an  arbitrary  constant  to 
the  result,  for  the  integral  of  the  proposed  differential 

Thus,  the  integral  of 

(a  +  ic"  +  y"'y-'^  X  -p {nx''-^dx  +  my'^-'^dy) 
c 

is  easily  seen  to  be  expressed  by 

h 


c 


(a  +  a?"  4-  y'^y  +  c, 


and  that  of        C^xfdFx    is     5^^' _|.  c 

(Young's  "  Integral  Calculus,"  p.  2) ;  and  it  is  manifest  that 
the  integrals  of  all  the  differentials  to  which  we  have  re- 
ferred, can  be  found  by  the  preceding  rule. 

(2.)  The  integral  ^l^ll.  +  C  of  {fxfdYx,  admits  of  a 

transformation,  which  we  will  now  proceed  to  give.  Thus, 
representing  the  hyperbolic  logarithm  of  Fx  by  log  Fa?,  we 
get  from  the  exponential  theorem  or  formula  (J),  given  at 

p.  61,  {YxY^^=  1  +  (^  +  1)  log  Faj  4- 

(n  +  ly  (log  Fo^y      {n  +  1/  (log  l^x)'  , 

1.2  ^  1.2.3  +'*^' 

consequently,  ^    j_       +  C  is  easily  reduced  to 

1  .it:.  (ri  +  1)  (log  F«y  „ 

-^  +  log  Yx  +  ^ 1^— ^  +  &c.  +  C, 

or,  representing  +  C  by  C,  we  shall  have 


INTEGKAL  CALCULUS.  255 


71  +  1 


+  0 


for  the  required  transformation,  a  series  that  evidently  con- 
verges rapidly,  when  n  +  1  is  small.  If  ?i  +  1  =  0  or 
71  =:  —  i,  the  formula  reduces  to  log  ¥x  -j-  C\  which,  since 

71  =  —  1  is  the  integral  of  Fx~'^dFx  =  -^,  or  using  log  C  for 

the  arbitrary  constant,  and  writing  /  before  the  differential 

(according  to  custom)  to  indicate  its  integral  (the  /  being 
called  the  characteristic  of  integrals),  we  shall  have 

/dFx 
-.g^—  =;  log  Fa? + log  C  =  (from  the  nature  of  log.)  log  CFa?. 

Hence,  the  integral  of  the  differential  of  a  function  divided 
by  the  function,  can  be  found  by  the  following 

RULE. 

The  integral  of  the  differential  of  a  function  divided  by 
the  function,  equals  the  hyperbolic  logarithm  of  the  function 
plus  an  arbitrary  constant;  or,  which  comes  to  the  same, 
the  integral  equals  the  hyperbolic  logarithm  of  the  product 
of  the  function  and  an  arbitrary  constant 

Eemark. — Any  constant  factor  (or  divisor)  of  the  differ- 
ential, must  be  retained  in  the  integral,  or  the  integral  must 
be  multiplied  (or  divided)  by  it,  according  to  the  case. 

The  rule  here  given  is  clearly  the  reverse  of  that  given  at 
p.  54  (when  77i  the  modulus  =  1),  for  finding  the  differential  of 
the  hyperbolic  logarithm  of  any  expression ;  deduced  from 

d  (log  x)  =  —       or   from    d  (log  ¥x)  =  -^pr-  • 


256  INTEGRAL  CALCULUS. 

Hence,  the  integrals  of  the  differentials,  found  under  the 
rule  at  p.  54,  will  reproduce  the  examples,  after  thej  are  cor- 
rected by  the  introduction  of  the  requisite  constants. 

Thus,  /5|±|'  =  log(a,  +  y)  +  C, 

as  in  example  2,  and 

/%nxdx  r  2xdx  ,       /  2  ,   ^\    ,    n 

which,  if  m  is  the  modulus  of  common  logarithms,  is  equiva- 
lent to  log  (a*  +  a?^)  by  using  Log  before  a^  +  ar^  to  express  its 
common  logarithm ;  as  in  example  3  (see  p.  55). 

rp,     .   ,        ,      ^    dx  dx  ,        dx  dx 

The  mteffi-als  of and 1 are 

°  a  +  x      a  —  x  x -\- a      x  —  a 

log  {a  +  x)  +  log  (a  —  £P)  +  C  =  log  {a"  —  a^)  +  C, 

and    log  {x  +  a)  +  log  (x  —  a)  +  log  C  =  log  C  (ar*  —  a/). 

The  integrals  of ^  and  -s — ^r^, ,  when  reduced  to 

°  a  +  2x  a^  -f  3ar ' 

proper  forms,  are 

by  the  nature  of  logarithms,  and 
Qxdx 


-f- 


The  integral  of 
dx 


=  iogG{d^  +  s^y 


/dx 


INTEGRAL  CALCULUS.  257 

(3.)  From  wliat  is  shown  in  (9),  at  p.  72,  the  following 
table,  which  is  very  useful  in  finding  integrals  of  differen- 
tials of  certain  forms,  by  means  of  circular  arcs  to  radius 
unity,  is  easily  seen  to  be  correct. 


-77-2 — ^IZS■^  =  sm-^-a?  +  C, 

/hdx  .h      ,  ^ 

-77-2 — jr:2{  =  cos-^-a?  +  0, 
|/  (or  —  trar)  a 


/obi 


Jo' 


,2  „  _:  tan-^-a;+  0, 
6V  a 


ahdx  ^    ,h      .   ^ 

=  cot-^-  »+  C, 


-/ 
/ 


+  &V  a 


adx  1 5       ,   ^ 

sec  ~  ^  -  a?  +  U, 


ac/a?  1  ^.  /M 

/  /^- 2^  =  c*^sec  -  ^  -  a?  +  C, 

a?|/(5V  — a^)  a 


hdx  _         .   _i  25= 

4/  (a^a?  —  J^a?^)  ~  a 


=  versin-^  -72"^  +  ^) 


r      hdx  .    .w    ,  ^ 

-  y   ^(a-a.-5V)  =  ^^^^'^^^-'  ^  ^  +  C- 

In  using  this  table,  it  must  be  observed,  that  by  the 

notation  sin*"-^-  a?,  is  meant  an  arc  of  a  circle  whose  radius  is 
a 

unity,  and  sine  -  a?,  and,  in  like  manner,  the  remaining  ex- 
pressions are  to  be  understood.  (See  Young's  "Integral 
Calculus,"  p.  10,  and  p.  20  of  his  "Differential  Calculus.") 


258  INTEGRAL  CALCULUS. 

To  perceive  the  use  of  the  table,  take  the  following 

EXAMPLES. 

1.  To  find  the  integrals  of     ,,'- sr    and    ——■ -r  . 

Putting  ic^  =  y,   the    second   of   these    forms    becomes 


.,,  ,  which  is  similar  to  the  first  form.     Hence,  since 

here  a  =  1     and     ^  =  1,  the  integrals  will  be  expressed, 
according  to  the  first  forra  of  the  table,  bj 

dx 


I 


V(i-V) 


sin-^  a?  +  0 


and          \—7Tr- — Tx  =  sin-^  v  +  C  =  sin-^  ar^  +  C. 

2.  To  find  the  integrals  of 

dx  ,  dx 

and 


^{\--^x^)  |/(4-ie-)' 

Bj  putting  1  and  2  for  a  and  h  in  the  fii*st,  and  the  reverse 
in  the  second  form  of  the  table,  we  readily  get  the  integrals 
expressed  by 


-M- 


2dx  1  in        ^ 


|/(l-4^'-)      2 

and  —  f-jTi-—^  =  cos  -^  ^  +  C. 

J   \/{4l  —  x-)  2 

3.  To  find  the  inteerals  of  - — '■ — 5     and     ± 77-5 . 

°  1  +  0?-  4  -h  9ar 

Putting  1  for  a  and  for  h  in  the  first  of  these,  and  2  and  3 
for  them  in  the  second,  we  get,  from  the  third  and  fourth 
forms  of  the  table,  the  integrals 

r    dx         ^       1      ,   n         ,     ^    r    ^dx         tan-' 3      ,  ^ 
•/  1  +  ar  ^4  +  93?-^      cot"^  2 


INTEGRAL   CALCULUS.  259 


4.  To  find  the  integrals  of 


and 


X  |/  i^ar  -  4)  x^/  (9^-'  -  4) 

By  putting  2  and  3  for  a  and  h  in  the  fifth  and  sixth  forms 
of  the  table,  the  integrals  will  be  found  to  be 

J  X  ^  {'i)x^  —  4)  2 

1  r      Mx  i3    ,  n 

^^^  -y  719:^-4^  ="  "^^'^    2  ^ + ^- 

5.  To  find  the  integrals  of 

dx  T  dx 

-^     and 


i/{x  —  ar)  \/{4:X  —  \)x^)' 

Putting  a  =  l  and  5  =  1  in  the  first,  and  a  =  2  and  h  =  B 
in  the  second  of  these,  we  readily  get 

/•— 77 ^  =  versin-^  2x  -{-  G 
\/  (,'C  —  X-) 

,  1    r        Mx  1  .     ,9      ,   ^ 

6.  To  find  the  integral  of 

dx  ^  .,  .     ,     ^        a?    "  c?a? 

The  integral  of  the  first  form,  from  the  seventh  form  of  the 

table,  is 

r         dx  1  .     ,2h         ^ 

/  ~77 r^  =~/7:  versm-'  —x  +  G, 

J    |/  [ax  —  OX')       \/o  a 

and  the  integral  of  the  second  form,  from  the  first  form  of 
the  table,  is 

r_^ldx___^^(^^. 

J    i/{a-hx)~  i/b  Kal       '      - 

and  it  is  easy  to  perceive  that  these  integrals  are  equivalent 


260  INTEGRAL   CALCULUS. 

(4)  If  a  differential  consists  of  two  or  more  terms,  such 
that  it  has  an  algebraic  sum  of  factors  so  related  that  the 
differential  of  each  factor  is  multiplied  bj  the  product  of  all 
the  remaining  factors,  then,  from  reversing  the  rules  at  pp.  9 
and  10,  it  follows  that  the  product  of  all  the  (different)  factors, 
plus  an  arbitrary  constant,  will  be  the  integral  of  the  differential 

This  rule  is  easily  perceived  to  be  correct  by  reversing  the 
methods  of  finding  the  differentials  of  the  examples  at  pp.  9  to 
11.    Thus,  from  the  differentials  in  example  1,  at  p.  9,  we  have 

/  {xdy  4-  ydx)  =  xy  -\-  0 

and     r  {SMy  4-  Qyxdx)  =  3    /*  {a^dy  +  ydx")  =  Ssry  +  C. 
And  from  the  fifth  and  sixth  examples,  we  have 
a  C  {^xyHH^  H-  Ixz'ydy  -f  fzHx)  = 

a  j  {xyH^  +  xz^ay^  +  'i^^dx)  =  axy^z^  +  0, 
f{2xy~Hx  -  dg^y-'dy)  = 
f{y~'ds?  +  x'dy-')  =  a?V-'  +0  =  ^  +  0.     ^ 


and 


Also,  from  example  4,  at  p.  10,  we  have 

=  fi  -/(a'  +  ^)  X  d^{a'  -  x')  +  i/ia'^a^)  c^  ^/(a^  +  ar^)] 

=  |/(a'4-a?')i/(^'-aj2)  +  C=  ^  {a' -  x') -\- 0 ', 
and  in  the  same  way,  the  integral  of 

dx  y?dx 

X 

is  easily  found  to  be         ^        :   -}-  C  as  at  p.  10. 


INTEGRAL   CALCULUS.  261 

"Remarks. — 1.  Fi-om  the  first  of  these  examples,  we  have 
/  [xhj  +  yclx)  ==    /  xdy  +    /  ydx  =  a^y  +  C, 

which  gives  /  ijdx  —  xy  —  j  xdy  +  C, 

and  evidently  reduces  the  integration  of  ydx  to  that  of  xdy  ; 
and,  in  like  manner,  the  integral  of  xdy  is  reducible  to  that 
of  ydx.  This  process,  which  is  often  very  useful  in  finding 
and  simplifying  integrals,  is  called  integration  hy  parts. 

2.  To  illustrate  this  method,  we  will  apply  it  to  find  the 
integral  of  Xc/a?,  supposing  X  to  be  a  function  of  x. 

Because  'X.dx  =  'X.dx  -f  xd^  —  xdJ^, 

by  taking  the  integrals  of  these  equals,  we  have 

fxdx=:Xx-  fxdX; 

and  since  dX.  =  -^  dx, 

dx      ' 

we  have  xdX.  =  —  xdx, 

.  .  ,     ,  r    ,^       dX  x"        rdPX  x'dx 

whichgives  JxdX=    ^  ^  -J-^  — . 

k     -t  »    •^•^  n         I  a  -A.  X  ax 

And  m  like  manner,  from  /  -^-^  ^^r- 
J   dor     2 

,  d^X  x^         r  „^  ^dx 

^^^^^^  ^2:3 -7°^^  273-' 

and  SO  on,  to  any  extent  required. 

Hence,  from  the  substitution  of  these  values  in 


/  Xdx  =  Xx  —  I  -J-  ^dx^ 


we  get 

/^  7      ^        dX  x^       d'X    X?        d^X      X*  .        ^ 


262  INTEGRAL  CALCULUS 

which  is  called  the  Formula^  or  Series,  of  John  BetmouilU  ; 
which  clearly  shows  that  the  proposed  integral  can  always  be 
found,  at  least  in  a  series. 

If  we  put  X,  x^j  ar*,  &c.,  successively  for  X,  in  the  formula, 
we  shall  get 

a^      ^      or 


f 


xdx  =  a^-  ^~  4-C  =y  +  C, 

and  so  on  ;  results  which  are  evidently  correct. 

It  may  be  added,  that  Maclaurin's  Theorem  is  applicable 

to  the  expansion  of    /  Xdx.     Thus,  putting  /  Xdx  for  X  in 

Maclaurin's  Theorem  (J),  given  at  p.  17,  we  shall  have 

fx,.=  ifx^HiX).  +  (f )  ,^  +  (g)^  +,&., 

for  the  required  expansion ;  in  which,  for  x,  we  must  put 
naught  in  the  expressions  within  the  parentheses,  and  the 

term  (  /  'X.dx)  must  clearly  represent  the  arbitrary  constant 
Thus,  if  we  put  a^  for  X,  the  formula  becomes 

fx'dx  =  C  +  l'  ; 

since  (af),  (Saf),  and  (3-i?2.2j),  are  reduced  to  naught,  when 
naught  is  put  for  x  in  them,  while  the  term 

(6^X\      x*  x^  a* 

17}  0:3.1    ^"*=°"^=^    3  X  2  X 1  X  j-2-3-^  =  ^. 

Mr.  Young  (at  p.  81  of  his  "  Integi'al  Calculus")  says,  that 
Maclaurin's  Theorem  fails  to  be  applicable,  when  x  =  0 
reduces  the  preceding  coefficients  to  naught ;  which  is  cer- 


INTEGEAL   CALCULUS  263 

tainly  incorrect,  since  tlie  Theorem  is  always  applicable 
when  it  has  no  infinite  term  (see  p.  17).  It  may  be  added, 
that  Maclaurin's  Theorem  is  (generally)  more  nseful  in 
practice  than  that  of  Bernouilli. 

(5.)  By  reversing  the  rule  at  p.  66 j  we  easily  find  the 
integral  of  an  exponential  differential,  when  the  exponent 
of  the  exponential  is  alone  variable,  by  the  following 

RULE. 

Divide  the  differential  by  the  hyperbolic  logarithm  of  the 
base  of  the  exponential  and  by  the  differential  of  its  expo- 
nent, and  add  an  arbitrary  constant  to  the  result  for  the  integral. 

The  truth  of  the  rule  is  manifest  by  revei'sing  the  method 
of  finding  the  differentials  at  p.  56. 

Thus,  from  the  first  example  we  have 

y*2-^  log  2dx  =  1^  log  2  dx  -^  log  2  (^a?  +  C  ==  2^  +  C, 
and  Jzy  log  3  c^y  =  3^  +  C. 

Also    /  a^  log  adx  =  a^  +  C    and      /  e^dx   ==  e^  +  C, 

supposing  e  to  be  the  hyperbolic  base. 

And  from  reversing  the  rule  at  p.  59,  it  results  that  the 
integral  of  a  cosine  of  a  variable  multiplied  by  the  differen- 
tial of  the  variable,  equals  the  sine  of  the  variable  plus  a 
constant.  Also,  the  integral  of  minus  the  sine  of  a  variable 
multiplied  by  the  differential  of  the  variable,  equals  the  cosine 
plus  a  constant 

Thus,  /  cos  1x  X  2dx  =  sin  2a?  +  C, 
and  —  /  sin  Zx  x  Zdx  =  cos  Sx  +  C. 
And     /  cos  4:xdx  =  -    I  4:dx  x  cos  4.-^?  =  — j 1-  C,   and 


264  INTEGRAL  CALCULUS. 

sin  6x  X  2dx  =  —  -   I  sin  5j;  x  5dx  =  -  cos  ox  +  C, 

and  so  on,  as  in  reversing  the  examples  at  p.  60. 

Kemarks. — We  might,  in  like  manner,  proceed  to  reverse 
the  rules  at  p.  60,  &a,  but  we  do  not  think  it  necessary  to 
consider  them  any  further  in  this  place. 

(6.)  If  any  number  of  differentials  are  connected  together 
by  the  signs  +  and  — ,  it  is  manifest  that  they  are  to  be 
considered  as  constituting  one  differential,  whose  integral 
requires  only  one  arbitrary  constant. 

Thus,  if  we  have 

2axdx  +  Sha^dx  —  4:ca^dx, 
it  is  to  be  considered  as  a  single  differential,  having 

/  {a2xdx  +  5  X  Sa^dx  —ex  4a^dx)  =  aa^  -}- hx^  —  ex*  +  G 

for  its  integral,  C  being  the  arbitrary  constant    Keciprocally, 
for  any  expression  like 

/  {oix^dx  +  P  X  dx  —px^dx  +,  &c.), 

we  may,  if  we  please,  write 

a  J  Mx  4-  bj  Mx  —J>J  x^dx 

or  J  {aa^dx  +  hx^dx)  —J  ^dx,  &c. 

(7.)  It  may  be  added,  that  the  arbitrary  constants  in  in- 
tegrals, are  (generally)  to  be  determined  so  as  to  satisfy 
certain  conditions  which  the  integrals  must  answer. 

Thus,  if  the  integral  /  {Sx-dx  +  6x*dx)  must  equal  naught 
when  X  =  a,  we  proceed  as  follows.     By  integration  we  have 
fiSx'dx  +  5x'dx)  =  x'  ■i-x'  +  C] 


IJ^EGRAL   CALCULUS.  265 

consequently,  putting  a  for  x  in  this,  we  must,  from  the  con- 
ditions of  the  question,  have  ^-^  +  a^  +  C  =  0,  which  gives 
G  =  —  {a^  -\-  aF)  for  the  value  of  the  constant  Hence  the 
integral,  duly  corrected,  becomes 


/( 


{^x'dx  +  bx'dx)  =  u?  -\-x'—  {w"  +  a') ; 

and  it  is  manifest  that  we  must  proceed  in  like  manner  in  all 
analogous  cases. 

To  signify  that  an  integral,  as 

/  {axdx  +  hfdx  —  ca^dx) 

is  to  be  taken  from  x  =  A  to  a?  =  B,  we  write 

/{axdx  -f-  hx^dx  —  co^dx)  =  -^  -\-  — 7-  +  ^» 
A                                                                     Z             o             4: 

which,  by  putting  A  for  a?,  gives 

^  (aA'      hA'      cA'\ 

^  =  ~v^~^"3 — r)^ 

and  thence  the  integral  becomes 

{axdx  +  bx'dx  —  ca^dx) 

~2'^3         4        V2"^    3  4"r 

which,  by  putting  B  for  aj  in  its  right  member,  becomes 

/    {axdx  +  ha^dx  —  ca^dx) 

oB^      hW_cB^_  /oA^      hA^  _  cA*\ 
2    "^   3         4         I  2    "^"    3~      ~T) 

=  |(B^-A^)-f|(B3-A>^)-|(B^-A0, 

which  is  called  a  definite  integral^  because  a?,  in  its  right 
member,  is  determined;  consequently,  when  an  integral  is 

12  .  _     * 


/: 


266  INTEGRAL  CALCULUS. 

taken  from  one  value  of  its  variable  to  another  value  of  its 
variable,  the  integral  is  d^'fiiiite  or  detecmined^  otherwise  the 
integral  is  indefinite  or  not  fixed, 

(8.)  When  the  integral  of  a  proposed  differential  is  fcund, 
it  is  said  to  he  integrated ;  and  when  the  integral  is  taken 
from  one  value  of  the  variable  (a?)  to  any  other  proposed 
value,  it  is  said  to  be  integrated  from  the  first  to  the  second 
value  of  the  variable. 

(9.)  To  aid  in  what  is  to  follow,  and  to  show  the  natures 
of  differentials  and  integrals  more  fully,  we  will  now  pro- 
ceed to  give  the  solution  of  the  following  important 

PROBLEM. 

If  a;  and  y  —fix)  =  a  function  of  a?,  represent  the  abscissa 
imd  corresponding  rectangular  ordinate  of  a  plane  curve,  it 
is  proposed  to  show  how  to  find  the  area  bounded  by  the 
ordinate  drawn  through  the  origin  of  the  co-ordinates,  by 
any  other  ordinate,  and  the  intercepted  parts  of  the  axis  of  x 
and  the  curve :  supposing  the  ordinate  to  be  constantly  posi- 
tive between  the  preceding  limits. 

It  is  clear  that  we  may  suppose  f{x)  to  be  expressed  by 
A  -h  Baj«  +  C^*  +  Dxf"  +,  &c., 
in  which  A,  B,  C,  &c.,  a,  J,  c,  &c.,  are  independent  of  a?, 
which,  for  simplicity,  we  shall  suppose  to  be  positive,  and 
that  a?",  a?*,  «*',  &c.,  are  arranged  according  to  the  ascending 
powers  of  x. 

Let  then,  in  the  figure,  0  be  the  origin  of  the  co-ordinates, 
and  suppose  04  represents  any  abscissa,  and  4^  the  corre- 
sponding ordinate ;  we  propose  to  find  the  area  or  quadrature 
of  the  curve,  bounded  by  the  ordinates  Oa  and  4e  =  y^  the 
abscissa  04  =  a?,  and  the  portion  of  the  curve  ae. 

Suppose  a?  to  be  divided  into  any  number  {n)  of  equal 


PEOBLEAf. 


2G7 


parts  at  the  points  1,  2,  3,  &c.,  and  let  oc'  represent  any  one 
of  these  parts;  then,  the  ordinates  corresponding  to  the 
points  0,  1,  2,  3,  &c.,  may  evidently  be  expressed  by 

A  =  y\    A  -1-  Ba?'«  +  Qx">  +  &c.  =  y", 

A  +  B  {^xj  +  C  (2^7  +  &c.  =  y'", 
and  so  on  to 

A  +  B  {nxj  +  C  (?i£t'7  +  &c.  ==  2/"'''- 
Allowing  the  rectangles  to  be  drawn  as  in  the  figure,  it  is 
easy  to  perceive  that  the  sum  of  all  the  inscribed  rectangles 
will  be  expressed  by 

(y'  +  2/^'  4-  . . .  .  +  y"")  x'  = 

Anx'  +  B  [1  +  2"  +  3«  + +  {n  —  1)^]  x"'^'^ 

+  C  [1  +  2*  +  3*  +  . . . .  +  (n  -  1/]  x"^^  +,  &c., 
as  is  manifest  from  the  principles  of  mensuration,  while  the 
sum  of  all  the  corresponding  circumscribed  rectangles  will 
be  expressed  by 


268  INTEGRAL  CALCULUS. 

(/'  +  /"  +  ....  +2/""0«''  = 

Knx'  +  B  [I  +  2-^  +  3«  +  ....  +  ^T  a;'°+^ 

+  C  [1  +  2*  +  3*  +  . . . .  +  n*]  x"'^^  +,  &c. 

It  is  easy  to  perceive  that  the  difiference  between  the  pre- 
ceding sums  of  the  circumscribed  and  inscribed  rectangles  is 
expressed  by 
(y"+^-y>'=B;iV +H  C/iV*+^  4- &c.  =  (B.r«  +  C.r'' +  &c.)  aj', 

since  nx'  =  x.  If  x'  is  unlimitedly  small,  the  difference  is 
evidently  also  unlimitedly  small ;  consequently,  since  the 
difference  is  clearly  greater  than  the  difference  between  the 
sought  area  of  the  curve  and  the  sum  of  all  the  inscribed  or 
circumscribed  rectangles,  it  is  manifest  that,  by  taking  x' 
sufficiently  small,  the  sum  of  all  the  inscribed  or  circum- 
scribed rectangles  may  be  made  to  differ  from  the  sought 
area  of  the  curve  by  a  difference  which  shall  be  unlimitedly 
small.    (See  Lemma  IL,  Book  L,  of  Newton's  "  Principia.") 

It  is  clear  that  what  has  been  done  holds  good,  whether  ae 
is  a  curve  or  straight  line,  or  even  if  it  is  a  curv^e  whose  con- 
vexity is  turned  toward  the  line  of  the  abscissae  or  the  axis  of  x. 

We  now  propose  to  put  the  above  expressions  for  the 
sums  of  the  inscribed  and  circumscribed  rectangles  under 
more  useful  forms. 

By  putting  n  —  1  =  ti',  it  is  clear  that  we  may  assume 

1  +  2«  +  3«  +  . . . .  4-  n'-  =  P^i"^-^^  +  Q/i'"  +  'Rn'--'  +,  kc, 

and  suppose  P,  Q,  &c.,  to  be  independent  of  n'   and 

1  +  2^4- 3*+,  &c., 

clearly  admit  of  like  representations.  By  changing  n'  into 
7i'  +  1,  and  subtracting  the  assumed  equations  from  the 
results,  we  get  the  identical  equations 


PROBLEM.  269 

P  [(a  +  1)  W<^  +  (^l)f  ^/a-i  +  &c.]  + 

a.  [an^«-i  +  ^(^"Zi)  ^,'a-2  4_  &c.]  +,  &c., 

and  so  on;    of  course,  by  equating  the  coefficients  of  like 
powers  of  n' ^  in  the  members  of  the  equations,  we  readily  get 


~«  +  l'   ^~2'  8.4'  '  1.2.8.4.5.6     ' 

&c. ;  and  so  on,  for  the  other  representations. 

From  the  substitution  of  the  preceding  values  of  P,  Q,  &c., 
by  putting  for  n'  its  value  n  —  1,  and  expanding  the  powers 
of  72.  —  1  according  to  the  descending  powers  of  7i  (as  hereto- 
fore) by  the  binomial  theorem,  we  get 

1  +  2«  +  3«  +  ...  +  {n-ir  -  ^^  +  ^.^  +  &c. 
^  a-\-l  1.2 

^a+l    ^     ^a         ^^^a-1  ^  (^  _  1)  (^  _  2)  n«-l 


a  +  1       1.2   '     8.4  1.2.8.4.5.6 

and  by  changing  a  into  J,  6^,  &c.,  we  get  the  corresponding 
representations  of  1  +  2*  +  3*  +  . . .  +  (ti  —  1 )'',  and  so  on. 
It   may   be   proper   to    notice    here,   that    the    numbers 

Q  =  —  r-^ ,    R  =  — - ,  and  so  on,  called  the  numbers  of 

l.Z  {5.4: 

(James)  Bernouilli,  may  easily  be  calculated  to  any  extent,  by 
solving  the  equations 

"^  1.2  "^  1.2.3  ^  1.2.3.4  ~    ' 
and  so  on.     (See  p.  98,  Vol.  Ill,  of  Lacroix's  "Traite  du 
Calcul  Differentiel,"  etc.) 


270  INTEGRAL  CALCULUS. 

Hence,  from  the  substitution  of  the  preceding  values  in 
the  expression  for  the  inscribed  rectangles,  at  p.  267,  we 
shall  have ' 

Ar'  1  1 

tI  "  O  f ^  +  ^  ^''^')"  +  ^  ^^"^'^^  +  ^'-^  '''  -^%A 

[aB  {r^y-^  +  IQ  {nxj-^  +  &c.]  x-  -  y:^^^ 

\a{a-l)  {a-2)B(nx'y-^  ^Jj{h-1)  (b  -  2)  C  {nxy-'  +  kc] 

x'^  +  &;c.  =  (since  nx'  =  x)  Ax  -\ ^  +  -^ — -  +  &c. 

^  ^  a  +  1         h-\-l 

[aBx<^-'  +  hCx'-'  +  &c.]  x'^  -  y;^jjq 

[a{a-l){a-2yBc(f-'  +  h{b-l){h-2)Cx'-'  +  kc.']x''  +  ,kG- 
'  If,  see  p.  268,  (B:»«  +  C^*  +  &c.)  x'  is  added  to  the  right 
member  of  this  equation,  the  sum  will  express  the  circum- 
scribed rectangles,  and  we  shall  have 

(y"  +  y'"  +  ....  +  r*')  x'=.Ax  +  ^^  +  "f-^  +  &o. 
Ay  1 

.'  +  i^  [aB.«-  +  hCx^-^  +  &c.]  .'^  -  j^L__ 

[a(a-l)(a-2)B^^-3-f2>(?y-l)(7>-2)0.iJ*-3+&c.] .»''  +  ,  &c. 
It  is  easy  to  perceive  that  the  part 


6  +  1 


Baj'^  +  1       Cx' 
of  the  inscribed  an  1  circumscribed  rectangles,  which  is  inde- 


PROBLEM.  271 

pendent  of  x\  or  does  not  depend  on  tlie  number  of  equal 
parts  into  whicli  04  ==  a?  is  supposed  to  be  divided,  must 
express  the  area  of  the  curve  bounded  by  the  ordinates  Oa 
and  4^^,  and  the  intercepted  parts  04  and  ae  of  the  line  of  the 
abscissae  and  curve,  as  required ;  it  is  also  evident  that  the 
terras  in  the  rectangles,  which  involve  x'  and  its  powers  as 
factors,  must  depend  on  the  number  of  equal  parts  into 
which  04  =  a?  is  supposed  to  be  divided. 

T^  ,      ,       Bx^nx'       Oxhix' 

J^rom  Ana?  + — \-  -^ q-  +,&c., 

(X  +  1  />  +  1 

which  is  the  first  form  of 

.       ,   Ba;«+i    ,    Qx'^\    ,    - 

it  is  clear  that 

(A  +  Bx"  -f  Ce»^  +  &c.)  x'  =:f{x)  x'  ==  yx' 

is  equivalent  to  the  differential  of  the  curvilinear  area  046a, 
and  may  be  expressed  by  writing  dx  for  x' ;  noticing  that 
the  x'  here  used  need  not  be  the  same  as  the  x'  in  the  other 
terms  of  the  rectangles  described  above.  Also,  multiplying 
by  n,  and  putting  nx'  ==  ndx  ■=  a?,  which  gives 

is  clearly  the  same  as  the  integral  of  the  preceding  differen- 
tial, since  the  results  are  found  by  measuring  the  index  of  x 
in  each  term  of  the  diiferential  by  unity  or  1,  and  dividing  by 
the  measured  index  of  a?,  which  is  in  conformity  to  the  com- 
mon rule  for  finding  the  integral  of  the  differential  of  a  power. 

It  is  hence  clear  that  the  Differential  and  Integral  Calculus 
are  deducible  from  what  has  been  done,  without  using  infin- 
itesimals or  limiting  ratios.     [See  (17)  at  p.  44.] 

It  is  hence  easy  to  perceive  in  what  sense  the  Integral 


272  INTEGRAL  CALCULUS. 

Calculus  may  be  regarded  as  being  tbe  reverse  of  the  Differ- 
eatial  Calculus,  and  vice  versa. 

Representing  A  +  Ba^  -h  Ca?*  +,  &a,  by  y,  tbe  expression 
for  tbe  sum  of  the  inscribed  rectangle,  becomes 
(y'  +  S^'' +  ....  + y")aj'  = 

Jy<^  +  -^  -  ^2  +  3.4  ^  ^  1.2.3.4.6.6  c^a^  ^  +'  ^''- ' 
since 

aB.'-'  +  50,.-  +  &c.  =  ^(A  +  B.°  +  &cO  ^  I  _ 

and 

[a  (a-1)  (a-2)  Bir«-H  J  (J-1)  (5-2)  C»*-'+  &c.]  = 

^^^  [A  +  Baj«  +  Caj*  +  &c.]  -f-  6/aj«  =  ^^  , 

and  so  on ;  a  form  which  is  substantially  the  same  as  given  by 

Lacroix,  at  p.  107  of  his  work,  from  very  different  principles. 

By  adding  {y—y')  x'  to  the  right  member  of  the  preceding 

equation,  we  shall  have  {y"  -\-  y'"  + +  y)  «?',  the  sum  of 

the  circumscribed  rectangles,  expressed  by 

It  may  be  added,  that  in  the  inscribed  rectangles  y'  is  the 
first  ordinate  and  y^  the  last,  while  in  the  circumscribed  rect- 
angles y"  and  ]p  +  ^  are  the  first  and  last  ordinates. 

If  the  preceding  equations  are  divided  by  x\  and  ly  is 
used  to  express  the  sum  of  the  ordinates,  taken  according  to 
the  preceding  directions,  we  shall  have 


Jydx 


y'     2/1  dy  ^>        1      ^y 


^y~     x'     ■^1.2     1.2  "^3.4^2^ ^~1.2.3.4.5.6c^^^""^'^''-' 
and      r 

ly  -ill  ^  VL  j^    y^  j^  1.^^' ^L___^'3,    ^c . 

•^  ~     x'  1.2  +  1.2  +  3.4  dx""       1.2.3.4.6.6      ^'  ^^' ' 


PROBLEM.  273 

if  we  add  y  to  tlie  members  of  the  first  of  tliese  equations 
and  y'  to  tliose  of  the  second,  and  write  S  before  y  to  signify 
the  sum  of  all  the  ordinates,  then  the  two  equations  concur 
in  giving 
^y^Iy^y 

-     x'      ^      "l      ^ZAdx"^       1.2.3.4.5.6  ^^-^      +'^'^- 
This  formula  enables  us  to  find  the  exact  or  approximate 
values  of  series,  whose  terms  follow  a  given  law  of  forma- 
tion, and  are  equidistant  from  each  other,  or  have  equal  in- 
tervals between  them. 

Thus,  to  find  the  sum  of  the  series  1^  +  2'^  +  3'  +  ....+  a?^ 
we  have  y  ■=  x^,  called  the  general  term  of  the  series. 

Hence,       Jyd,e  =  ^-\-C,     J  =  2a?,     and     ^|  =  0, 

and  since  the  difference  of  the  successive  terms  of  the  series 
0,  1,  2,  3,  &c.,  equals  1,  we  put  1  for  x',  and  ^y  becomes 

— S"  +  ^^  +  C,  the  arbitrary  constant  C  being  =  0  since 

the  value  of  y\  which  corresponds  to  0  in  the  series  0, 1,  2,  3, 
&c.,  is  equal  to  0 ;  consequently,  %  is  reduced  to 


X 

3         2     '   2^3 


+    .0, 


to  which,  adding  y  =  x^,  we  have 

_  _  _  _  , 

for  the  sum  of  x  terms  of  the  proposed  series.     In  like  man- 
ner, to  find  the  sum  of  the  series  1,  2^,  3^ ar',  we  have 

12* 


274  INTEGRAL  CALCULUS. 

y'  —  0  and  y  =  xr^^  and  thence 


^«  i  +  0,  ..a  |.  W,     g=  1.2.3,   Jj  =  0,  te 


/ 

Hence,  the  formula 

-J    X'    ^     2      +3.4^a."^       1.2.3.4.5.6  c/^'^  ^    +'*''•' 

since     x'  =  1,    gives     %  =  ^  +  2    "^  ^  ~  120  "^  ^' 
in  which  C  is  the  arbitrary  constant     To  determine  C,  since 
Sy=0  when  x=  0,  by  patting  a?  =  0  we  get  0  =  —  j^  +  C, 

which  gives  C  =  zr^-z  ;  consequently,  we  shall  have 

for  the  sum  of  x  terms  of  the  proposed  series. 

(10.)  It  clearly  follows,  from  what  has  been  done,  that  the 
diiferential  of  a  function  of  a  single  variable  as  x,  being  of  the 
form  f  (x)  dx,  by  putting  fx  =  y  becomes  f  (a?)  dx  =  ydx ; 
which  may,  if  we  please,  represent  the  differential  of  the  area 
of  a  plane  curve,  whose  ordinate  corresponding  to  the  ab- 
scissa .T,  is  represented  hy  y  =f  [x). 

Thus  (see  the  fig  at  p.  267),  if  Sd  =  y  —f{x)  and  3, 4  =  dx^ 
the  product  ydx  =zf{x)  dx  =  the  area  of  the  rectangle  3(^D4, 
which  may  represent  the  differential  of  the  curvilinear  area  to 
the  right  of  Sd;  consequently,  the  area  to  the  right  of  the 
ordinate  Sd,  is  the  integral  of  the  differential,  supposing  it  to 
commence  at  the  point  where  the  curve  cuts  the  axis  of  x. 

If  ydx  =f  {x)  dx  is  the  differential  of  some  known  func- 
tion of  a?,  the  integral  /  /  (x)  dx  can  be  immediately  found ; 


PROBLEM.  275 

but  if/*(^)  dx,  can  not  readily  be  reduced  to  tbe  differential 

of  a  known  function  of  x^  then  j  f{x)  dx^    I  ydx^  being 

reduced  to  the  integral  of  the  differential  of  the  area  of  a 
curve,  will,  from  the  sum  of  the  inscribed  rectangles,  given 
at  p.  272,  become  (after  a  slight  reduction) 


/ 


ydx=.{^'  +  y"  +  ...,+y^)x'  _  g'  +  g  _  J^ 
dx"^    ^  1.2.8A5.6  (^^  '^''•' 


a  formula  that  will  enable  us  to  find  an  approximate  value 
of  the  proposed  integral,  when  x'  is  sufficiently  small,  par- 
ticularly when  taken  in  connection  with  the  integral 

fydx  =.  (/'+  y  "  +  .  •  •  +  2/  or  y"^')  ^'^  +  rl  -  H  -  ^ 

^.»^    +  1.2.8.4.5.6  (^^-^  '^'^•' 

deduced  from  the  formula,  given  at  p.  272,  for  the  sum  of  the 
circumscribed  rectangles. 

To  illustrate  what  is  here  said,  we  will  show  how  to  find 
dx 

T 

when  taken  from  the  limit  x  =  0  to  the  limit  x=l,  or  be- 

z — ^ — ^ .  [See  (7.)  at  p.  2  64.  ] 

Here  y  =  z ^ ,  which,  by  putting  x  =  0,  gives 

X  ~f"  X 

and  putting  a?  =  0.1  or  cc-  =  0.01,  gives 

?/'  =  ^-i^  =0  990099+; 


/dx 
'    ^  by  the  first  of  the  preceding  formulas, 
J.  -p  X 


276  INTEGRAL  CALCULUS. 

X  =  0.2,  gives  y'"  =  rSi  "^  0.961638+  ; 

and  so  on,  to  a?  =  0.9,  which  gives 

y^"  =0.552486+. 
By  adding  the  ordinates,  we  have 

2/'  + 2/'^  +  .... +2/^^=8.0998+; 
and  since  the  ordinates  are  drawn  at  intervals  of  0.1,  wo 
have  a?'  =  0.1 ,  and  hence  get 
(y'  +  2/''  +  .. . .  4-  2^10)  x'  =  8.0998  x  0.1  =  0.80998 +. 

Also,  _,|l^-=_0.05     and    ||' =  0.025, 

since  for  y  we  must  put  ^ ^  =  ^ ,  the  last  value  of  y. 

And  we  have 

dy  _  J       1  ,    _  2x 

Tx-"^' rr^ - ^^^ -  -  (iqr^y' 

which,  by  putting  1  (the  last  value  of  x)  for  a?,  gives 

^  _  _  1. 

dx~       2' 

consequently,       - -— ^  x'^  =  0.000416 +. 
0.4  cLx 

Hence,  rejecting  the  remaining  terms,  on  account  of  their  com- 
parative minuteness,  and  adding  the  terms  found,  we  have 

-^  =  0.80998-0.05  +  0.025  +  0.000416=0.78539+. 

ol  +  ar 

From  the  third  form  of  the  table  given  at  p.  257,  by  put- 

J  equals  the 

length  of  an  arc  of  45°  of  the  circumference  of  a  circle 


PROBLEM.  277 

whose  radius  =1,  wliicli  is  well  known  to  be  0.78539+  ; 
consequently,  tlie  arc  has  been  correctly  found  to  five  decimal 
places,  by  a  calculation  of  remarkable  simplicity.  By  draw- 
ing the  ordinates  sufficiently  near  each  other,  it  is  clear  that 
we  may  in  this  way  find  the  circumference  correctly  to  any 
finite  number  of  decimal  places. 

For  a  curve  such,  that  the  differential  of  its  area  is  that  of 
an  integral  of  a  known  form,  we  will  show  how  to  find  the 
area  of  a  parabola. 

Thus,  let  ax  =  y-  represent  the  equation  of  the  parabola ; 

then,  by  taking  the  differentials,  we  have  dx  =  ^,  which 
gives  ydx  —  — — ,  whose  integral  is 

from  the  equation  of  the  curve. 

To  find  the  constant  C,  we  shall  suppose  the  area  to  com- 
mence  at   the    vertex   of  the    curve;    then,   x  =  0  gives 

/  ydx  =:  0,  and,  of  course,  we  shall  have  0  =  0,  and  the 

area  becomes  /  ydx  =  -  xy  =  two-thirds  of  the  semi-par ah- 

olaJs  circmnscrihing  rectangle  ^  agreeably  to  a  vjeU-Jcnow7i 
j^roperty  of  the  parahola. 

(11.)  Eesuming  the  figure  at  p.  267,  and  supposing  it  to 
revolve  about  the  axis  of  x  or  04,  it  is  manifest  that  the 
curvilinear  area  will  describe  a  portion  of  a  solid  of  revolu- 
tion ;  and  that  the  inscribed  rectangles  will  describe  cylinders 
inscribed  within  the  solid,  while  the  circumscribed  rectangles 
will  describe  cylinders  circumscribing  the  solid,  such  that 
the  solid  will  be  greater  than  the  sum  of  all  the  inscribed 


278  INTEGRAL   CALCULtJS. 

cylinders,  and  less  than  the  sum  of  all  the  circamscribed 
cylindei*s. 

If  TT  =  3.14159,  &c.,  the  cylinder  generated  by  the  revolu- 
tion of  the  rectangle  01  Aa  will,  by  mensuration,  be  expressed 
by  "ny'-x'^  and  in  like  manner  all  the  remaining  inscribed 
cylinders  may  be  expressed. 

Hence,  if  y,  y\  y'\  &c.,  are  changed  into  ttz/^,  ttz/'-,  rry"^, 
&c.,  the  formula  for  the  sum  of  the  inscribed  rectangles,  at 
p.  275,  will  become 

Jiry^-cU  =  -nffdx  =  tt  \{:y"  +  y"'+  . . . .  +  2/"')  x'  -  Q^  -f 

1.2       3.4  dx   ^1.2.3.4.5.6  ^^         ^  ^^' 
the  formula  for  the  sum  of  all  the  cylinders  inscribed  in  the 
portion  of  the  solid  of  revolution ;  and  in  much  XhQ  same 
way,  the  sum  of  all  the  cylinders  which  circumscribe  the 
solid  may  also  be  found. 

Noticing,  that  this  process  will  be  unnecessary  when  the 

integral  expressed  by  /  y^dx  can  readily  be  found. 

Thus,  in  finding  the  contents  of  the  paraboloid  described 
by  the  revolution  of  the  parabola  ax  =  y^  about  the  axis 
of  X. 

Since        dx  =        ^ ,      we  have      iMx  =  -^-^ , 
and  thence  we  get 

which  equals  half  of  the  cylinder  which  circumscribes  the 
paraboloid ;  noticing,  that  no  constant  is  necessary,  since  the 
paraboloid  equals  naught  when  x  =  0. 

For  another  example,  we  will  show  how  to  find  the  con- 
tents {o?'  cuhature)  of  a  sphere  whose  radius  equals  R 


LENGTHS   OF   CURVES. 


279 


From  wliat  is  shown  at  pp.  210  and  211,  it  is  manifest  that 
if  the  sphere  is  cut  by  a  plane  whose  perpendicular  distance 
from  the  center  is  x^  the  section  will  be  a  circle,  such  that 
E-  —  »^  will  equal  the  square  of  the  radius  of  the  section ; 
consequently, 

7r  (R2  _  rji^^  dx^Tx  (:Wdx  -  xHx) 

is  clearly  the  differential  of  the  portion  of  the  sphere,  between 
the  cutting  plane  and  a  parallel  plane  passing  through  the 
center  of  the  sphere.     Hence,  by  taking  the  integral  from 


a?  =  0  to  a?  =  E,  we  have 


for  half  the  sphere ;  consequently,  the  contents  of  the  whole 
sphere  is  — -  E^,  which  clearly  equals  two-thirds  of  the  cir- 

o 

cumscribing  cylinder. 

(12.)  We  now  propose  to  show  how  to  find  the  lengths  of 
plane  curves. 


Thus,  let  AB  and  BC  represent  the  abscissa  and  ordinate 
of  any  plane  curve  AC,  having  A  for  its  vertex,  which  we 
shall  take  for  the  origin  of  the  co-ordinates,  supposed  to  be 
rectangular.  Then,  representing  the  arc  of  the  curve  AC 
by  2,  the  abscissa  AB  and  ordinate  BC  by  x  and  y,  wc  may 
clearly  take  the  very  short  line  Q>^  parallel  to  AB,  to  stand 


280  LENGTHS  OF  CURVES. 

for  dx^  the  differential  of  a?,  and  st  parallel  to  BC  or  y,  meet- 
ing the  tangent  to  the  curve  at  C  in  t^  to  stand  for  dy^  then 
it  is  clear  that  C/,  the  hypotenuse  of  the  right  triangle  C*^, 
must  equal  dz^  the  differential  of  2,  or  we  shall  have 

for  the  differential  of  the  arc  AC  =  z. 

What  is  here  affirmed,  is  clear  from  the  definition  of  a 
tangent  given  at  p.  125,  which  irs  that  the  differential  co- 
efficient -^  at  the  point  C  in  the  curve  must  be  the  same  as 

in  the  tangent;  consequently,  using  Gs  to  represent  dx^  st 
must  represent  dy^  and  thence  C^  must  clearly  represent  dz^ 
as  above. 

Because  the  approximate  method  of  finding  the  integral 
of  the  differential  is  sufficiently  evident  from  what  has  here- 
tofore been  done,  we  shall  not  stop  to  give  it. 

Thus,  to  find  the  length  of  the  curve  whose  equation  is 
y'  =  ax^^  called  the  equation  of  the  semicubical  parabola,  by 
taking  the  differentials,  we  readily  get 


dy=  -  a^x  ^dx, 
0 

and  thence 

dz  =  (x^-\-^a^\x~^dx) 

whose  integral  is 

.  =  (.*^^f+c, 

C  being  the  arbitrary  constant     If  x  and  z  equal  naught  at 
the  origin  of  the  co-ordinates,  we  shall  have 


(|a^)-fC  =  0, 


or    C  =  -^a, 


LENGTHS   OF   CUKVES.  281 


Hence  the  correct  integral  becomes 

consequently,  tlie  proposed  curve  is  said  to  be  exactly  recti- 
fi'Cible^  because  tbe  integral  of  its  differential  can  be  exactly 
found. 

For  another  example,  we  will  find  the  length  of  the  com- 
mon parabola,  its  equation  being  ax  =  y\ 

By  taking  the  differentials  of  its  members,  we  have 

adx  =  2ydy,     or    dx  =  ^— ^, 

2 

2^    2 

which,  by  putting  -^  =  h,  gives  dx^  —  ^-j~  ;  consequently, 


¥  +  v^ 
dx'^-h  dy''=       ./  dy% 


or  ^idx^  +  dy^)  =  dz  =  ^:^tt^ 

_        My  yHy 

Hence,  since 
and  that 


y'^dy         _  hdy 


we  have  reduced  -^-^ — ,      ^    '      to 


+  h:d.{lY  +  'yf, 


1^ 

2i 


282  LENGTHS  OF  CURVES. 

Hence  (see  the  last  example  at  p.  256),  we  stall  have 

=  I^^G^ +  *')  +  !  log  [y  +  4/(y' +  *^)] +c,     . 

by  representing  hyperbolic  logarithms  by  log,  and  using  C 
to  represent  the  arbitrary  constant. 

By  putting  y  =  0,  we   shall  clearly  have   2  -—  0,  and 

thence  C  =  —  ^  log  h ;  which  reduces  the  integral  to 

S^^'-^-l  ^^  +  *=)  +  ! log yA^^±Ii. 

Hence  the  common  parabola  is  rectifiable  in  algebraic  and 
transcendental  terms,  but  not  in  algebraic  quantities,  like  the 
preceding  example. 

For  another  example,  it  may  be  proposed  to  find  an  arc 
of  a  cycloid,  reckoned  from  its  vertex. 

By  referring  to  page  150,  we  have  dy  —  y dx, 

r  being  the  radius  of  the  generating  circle,  and  x  and  y  the 
abscissa  and  ordinate  ;  consequently, 

^  X         ' 


VWr  dx 
or. 


which  needs  no  correction,  supposing  the  integral  to  com- 
mence with  X. 

Hence,  see  the  fig.  at  p.  149,  it  is  clear  that  the  cycloidal 
arc  DG  =  2DF  =  twice  the  chord  of  the  corresponding  arc 
of  the  generating  circle ;  consequently,  DG^  or  z^  —  8rx, 


SURFACES   OF   SOLIDS   OF   REVOLUTIOIT.  283 

(13.)  We  will  now  proceed  to  show  how  to  find  the  sur- 
faces of  solids  of  revolution. 

Thus,  supposing  the  fig.  in  (12),  at  p.  279,  revolves  about 
its  axis  AB,  it  will  generate  what  is  called  a  solid  of  revolu- 
tion, whose  arc  AC  will  describe  its  curve  surfiice,  which 
we  propose  to  show  how  to  find. 

Because  Ct  —  (1.3  ^=  the  diiferential  of  AC  =  z,  it  is  mani- 
fest that  dz,  multiplied  by  the  circumference  of  the  circle 
whose  radius  equals  BC  —  y,  will  represent  the  differential 
of  the  surface  described  by  the  arc  AC  in  one  revolution 
about  its  axis  AB.  Hence,  putting  n  ==3.14159  +  ,  and 
representing  the  surface  described  by  AC  by  S,  we  shall 
clearly  have  dS  =  ^-nydz  for  the  differential  of  the  described 
surface. 

Thus,  to  find  the  surface  of  a  sphere  whose  radius  is  r,  we 
shall  evidently  have  r  \  y\',dz\  die  (or  CS),  (from  similarity  of 
triangles,  since  the  radius  drawn  to  C  cuts  C/^.perpendicularly, 
and  that  when  the  angle  TCB  is  acute,  tlie  center  is  at  the 
right  of  B  in  AB),  or  ydz  =^  7'dx ;  consequently,  c/S  =  ^rcydz 
reduces,  by  substitution,  to  c/S  =  lirrdx^  whose  integral   is 

I  dS  —   I  2Trrdx    or    S  =  27t/'^,  which  needs  no  correction, 

supposing  the  surface  S  to  commence  with  x.  If  for  x  we 
put  2/'.  the  integral  becomes  S'  =  4rr/'-,  where  S'  stands  for  the 
whole  surface ;  consequently,  since  nr  =  the  surface  of  a 
great  circle  of  the  sphere,  it  follows  that  S',  the  whole  surface, 
equals  four  times  the  area  of  a  great  circle  of  the  sjjhere  / 
and  from  S  =  ^^rx^  it  is  manifest  that  the  variations  of  S  are 
proportional  to  those  of  a?. 

If  we  take  dz^  in  the  parabola  given  at  p.  282,  we  shall 

have  ,S^?^f-+^y<^y, 


284  SURFACES  OF  SOLIDS  OF  REVOLUTION. 

for  the  differential  of  the  surface  of  the  comjnon  paraboloid, 
whose  integral 

fdS>=^f^{V'  +  f)ydy    gives   S  =  g  (J^  +  2^»  +  C, 

C  being  the  arbitrary  constant. 

To  determine  C,  we  suppose  S  and  y  to  commence  together, 
and  thence  get 

0  =  -~h\     which  gives   S  =  1^  [(5^  +  7/)^  -  J^ 

for  the  correct  integral. 

We  will  now  show  how  to  find  the  area  of  the  surface 
generated  by  the  revolution  of  the  catenarian  curve  about 
its  axis,  supposing  the  equation  between  the  length  of  the 
curve  and  the  corresponding  abscissa  to  be  expressed  by  the 
equation  z^=2ax-\-x^^  or  by  its  equivalent,  |/  (a-  +  2^)  =  a+x, 

By  taking  the  differentials,  we  have 

_        zdz 

consequently,  since  ds^  =  ds^  +  dy^,  we  have 

dj/  =  dz^  —  daf=  dr 2— — ■„  =  -j— — s 

^  a^  -h  s^       a^  +  ^ 

adz 

Because 

o?S  =  ^'nydz  =  27r  {^dz  +  zdy — zdy)  =  2  n-  {dyz  —  zdy), 
we  have,  by  taking  the  integral, 

C  being  the  arbitrary  constant,  which  equals  2na^  or 


SURFACES   OF   SOLIDS   OF   REVOLUTION.  28o 

when  S  =  0  at  the  vertex  of  the  curve.     Because 

our  equation  is  equivalent  to  S  =  2Tr  {yz  —  ax),  as  required. 

For  the  last  example,  we  will  show  how  to  find  the  surface 
generated  by  the  revolution  of  a  cycloid  around  its  base. 

Thus,  by  referring  to  the  fig.  at  p.  149,  since  BD  =  2r  and 
DE  =  37,  we  have  BE  =2r  —  x=^  the  perpendicular  from  Gr 
to  the  base  AC  of  the  cycloid,  and  which  revolves  about  the 
base ;  and,  fi-om  the  example  at  p.  282,  we  have 

(h  =  i/  —  dx  =  V  2r  x~^dx. 
^     X 

Hence,  putting  2r  —  x  for  y,  and  V  2rx~^dx  for  dz  in 
dS  =  2nydz,  we  shall  have 

c?S  =  2774/2?  (2r  —  x)  x~^dx  =  27TV2r(2rx~^dx  —  x^dx) 

for  the  differential  of  the  surface  generated  by  the  revolution 
of  the  cyclodial  arc  DG  about  BO,  since  this  increases  pos- 
itively, while  that  described  by  GC  decreases.  By  taking 
the  integrals,  we  have 

S:=27r  V2^{f2rx-^dx  -fx'^dx)  =2  tt  V2?  Urx^  ~\^% 

which  needs  no  correction,  supposing  the  integral  to  com- 
mence with  X.     By  putting  2r  for  a?,  we  have 

S=  277  V2^{4.r  \^-\r  V2?)  =  2r:V¥rx  \rV2^^=  ^^ 
for  the  surface  described  by  the  semicycloidal  arc  DC  about 
BC,  and  of  course  — ^  is  the  whole  surface  described  by 
the  revolution  of  the  cycloid  around  its  base,  as  required. 

(14.)  We  will  now  show  how  to  use  polar  co-ordinates  in 
finding  the  areas  and  lengths  of  curves. 


286 


USING  POLAR  CO-ORDINATES. 


Thus,  let  AC  be  <a  curvne,  having  P  for  its  pole,  PC  =  r 
and  the  angle  APC  =  </>  for  the  polar  co-ordinates  of  any 

point  C  of  the  curve ;  then,  shall  —^   equal  the  differential 

of  the  curvilinear  area  APC. 

For,  taking  PB  and  the  perpendicular  BC  for  the  rectan- 
gular co-ordinates  of  C,  and  denoting  them  by  x  and  y,  their 
origin  being  at  P ;  then,  from  what  has  been  shown,  ydbc  is 
the  differential  of  the  area  ABC.     (See  p.  266,  &c.) 

Also,  since  the  area  of  the  triangle  PBC  equals  -~^  and 

that  the  curvilinear  area  APC  =  the  area  ABC  —  triangle 

PBC  =  the  area  ABC  —  ^,  by  taking  the  differentials  of 

those  equals,  we  shall  have  the  differential  of  the  curvi- 
linear area 

APC  =  yd:c  -  W^A^  =  ydx_-xdy^ 
2  2 

Because      tan  angle  BPC  =  —  tan  <p  =  ~y 

by  taking  the  differentials  of  these  equals,  we  shall  have 

d(})  xdu  —  ydx  ,  ,         x^d(b         „ . 

TT-  =  — '■ — ^ — J     or     ydx  —  xdu  = — ^—  =  r^d(t> ; 

cos-<A  aj2        '  -^  -^      cos-^<A  ^' 

consequently,  we  have  the  differential  of  the  curvilinear  area 


USING   POLAR   CO-ORDINATES.  287 

APC  =  — ^r— ,  as  required.     Hence,  if  PE  makes  the  small 

angle  CPE  equal  to  c?0  with  CP,  d<^  being  an  arc  of  a  circle 
to  radius  1 ;  then,  it  is  clear  that  the  circular  sector  CPE, 
whose  center  is  at  P,  will  represent  the  differential  of  the 
area  APC. 

Thus,  if  AC  is  a  parabola,  having  Kx  for  its  axis"  and  P 
for  its  focus ;  then,  representing  AP  by  m,  4m  will  be  its 
parameter,  and  we  shall  have  (by  a  well-known  property 
of  the  curve) 

4m  {r)i  +  a?)  =  4m^  +  4m,'r  =  y^, 

or         Aiiri'  +  ^mx  +  a?^  =  (2  m  +  xf  =z  x^  -\-  y^  =i  r^^ 

which  gives  r  —  a?  =  2m  ;  or,  since  x  ^=  —  r  cos  0,  we  have 
r  {\  -\-  cos  </))  —  2m  ;  and,  since 


■777, 

1  +  COS  </)  =  2  cos^  J ,     we  get    r  = , 

cos^2 


2 


for  the  polar  equation  of  the  parabola. 


Hence,  — ^  becomes 
m-  -^       m^d .  tan  - 


cos*  -  COS^  J 


I-  (d.  tan  I  +  tan^  ^  d.  tan|j, 


whose  integral  taken  from  0  =  0,  gives 

as  required ;  a  result  that  is  very  important  in  treating  of  the 
parabolic  motion  of  comets.  (See  Yince's  "Astronomy," 
vol.  I.,  p.  428.) 

For  another  example,  we  will  find  the  area  of  the  spiral  of 


288  USING   POLAR   CO-ORDINATES. 

Archimedes,  whose  equation  is  r  =  a0.     By  taking  the  dif- 
ferentials, we  have 

,,        dr         1.1.  rH(l>       r^dr 

d<^=-,    which  gives    -^-  =^; 

whose  integral,  taken  from  ?'  =  0,  is 

/I'^dcp  _    7^ 

In  like  manner,  from  the  equation  r  =  a*,  the  equation  of 
the  logarithmic  spiral,  by  taking  the  hyperbolic  logarithms, 
we  have  log  r  =  <^  log  a,  whose  differentials  give 

^^       ^^  1                    ^^           ^^ 
—  =  a^  loff  a    or    d(b  =  —^ , 

r  °  r  log  a^ 

J  ^,  7'-o?0  rdr 

aad  thence  -tt-  =  ^r-, ; 

2  2  log  a 

whose  integral,  taken  from  r  =  0,  gives 

J     2     ~  4  log  a  ' 
By  taking  r  =  -  ,  or  0  =  - ,  the  equation  of  the  hyper- 
bolical spiral,  we  get 

,^            adr           ,  ^,             r'c?^            adr 
d(p= 5"  ,     and  thence     -^-  = p^- ; 

whose  integral,  taken  from  r  =  r\  is 

/7^d<\>  _  a  (/•'  —  r) 
~2"~~         2        ' 

which,  taken  to  r  =  0,  or  an  infinitesimal,  is 

/r'^d(^  _  ar' 

which  equals  the  area  of  a  right-angled  triangle,  whose  per- 
pendicular sides  are  a  and  r' . 


USING   POLAR   CO-OKDINATES.  289 

Remarks. — In  treating  of  spirals,  it  will  sometimes  be 
convenient  to  consider  the  pole  as  moving,  according  to  some 
given  law,  instead  of  being  fixed,  as  is  usually  done. 


Thus,  let  a  thread  be  wound  from  A  around  the  circle 
ADB  in  the  direction  of  the  letters  A,  D,  and  B;  then, 
when  the  thread  is  unwound  from  A,  so  as  to  be  constantly 
a  tangent  to  the  circle,  the  extremity  A  of  the  thread  will 
describe  a  curve  AC,  called  the  Involute  of  the  Circle^  to 
which  the  thread  is  clearly  constantly  perpendicular,  while 
it  unwinds ;  so  that  BO  denoting  any  unwound  part  of  the 
thread,  it  is  manifest  that  BC  cuts  the  curve  AC  perpendic- 
ularly at  C,  and  is  at  the  same  time  a  tangent  to  the  circle  at 
B,  and  equal  in  length  to  the  circular  arc  ADB.  (See  Sec. 
YL,  p.  163.) 

We  now  propose  to  show  how  to  find  the  area  of  the  invo- 
lute bounded  by  the  arc  AC,  the  unwound  part  CB  of  the 
thread,  and  the  circular  arc  ADB. 

Representing  OC  by  r^  the  radius  OB  of  the  circle  by  R, 
the  right  triangle  BOO  gives  BO  =:  |/  (/"^  —  R")  =  the  circu- 
lar arc  ADB.     Hence,  .^ '-  equals  the  arc  to  radius 

=  1,  which  represents  the  angle  AOB,  which  we  shall  take 
for  0,  and  for  r  we  shall  take  y  {f-  —  R-). 

13 


290  USING   POLAR  CO-ORDINATES. 

Hence,  from  (f>  =  -^ ,  we  get  #  =  gTT/^JZTRS) ' 

and  this  multiplied  by  t^  —  K^  tlie  square  of  the  correspond- 
ing radius  vector,  gives 

(/^  -  H')  d<f>  _  i/{7^-'R')rdr 
2  ~  2K 

for  the  differential  of  the  sought  area:  since  the  angular 
motion  of  BC  is  clearly  the  same  as  that  of  the  perpen- 
dicular radius  OB,  it  is  clear  that  tlie  angle  <{>  has  been  cor- 
rectly represented,  while  the  pole  moves  from  A,  on  the 
circular  arc  from  A  through  D  to  B. 

By  taking  the  integral  of  the  differential  equation,  we 

have  yii^__^_  =  k___L, 

for  the  correct  area ;  supposing  it  to  commence  wben  r  =  E, 
or  when  ^{?^—W)  =  0. 

Hence,  since  the  circular  sector  (fi'om  the  principles  of 
geometry)  ADB  =  OBC,  it  follows  tbat  we  shall  have  the 
area 

AOBC-OBC  =  the  area  AOC=  the  area  ACBD=  ^-g— » 

which  agrees  with  the  area  usually  found,  (See  p.  76  of 
Vince's  "Fluxions.") 

(15.)  To  find  the  lengths  of  curves  by  using  polar  co- 
ordinates, we  proceed  as  follows : 

Thus,  by  using  the  figure  and  notation  in  (14),  at  p.  285, 

we  have  x  =  —  7*  cos  ^    and    y  =  r  sin  (f>, 

which  give  dx=  —  cos  (fydr  +  r  sin  <pd(l> 

and  dy=  sin.  <pdr  +  r  cos  (pd(p ; 


LETTGTHS   BY   POLAR   CO-OKDINATES.  291 

wliicli,  by  taking  the   square  roots   of  the  sums  of  their 
squares,  gives 

^{dx^  +  dy-)  —  VT^d^^TdJ^  =  ^  Ir^  +  —^  X  d<}). 

From  what  is  shown  at  pp.  133  and  134,  since 
^{dx^  +  d>/^)  =  c?s,  the  differential  of  the  arc  AC, 
it  results  that  in  polar  co-ordinates  we  shall  have 

dB  =  V{rW  +  dr')  =  |/(/'^  +  ^')  # 

for  the  differential  of  the  arc  AC  =  2,  as  required. 

Remarks. — 1.  By  referring  to  the  figure  at  p.  131,  and  to 
what  has  there  been  done,  taken  in  connection  with  what 
has  been  done  above,  it  follows  that  the  normal  to  the  curve 
at  C,  limited  by  the  perpendicular  through  P  to  the  radius 

vector  PC  =  r,  equals  /(,^  +  '^,  since  (see  page  132) 
-—  =  the  square  of  the  subnormal.     Hence,  by  putting  the 

normal  y  1?'^  + -^-^l  =  N,   we  have   Nc?</),   from   what  is 

shown  above,  for  the  differential  of  the  curve  s,  in  polar  co- 
ordinates ;  in  which  (p  =:  the  angle  APC,  and  observing  that 
c?0  equals  the  differential  of  the  angle  which  the  perpendicu- 
lar to  PC  through  P  makes  with  the  axis  AB  of  x ;  noticing, 
that  (if  we  please)  we  may  regard  —  d(f)  as  being  the  differen- 
tial of  the  angle  which  the  normal  to  the  cutve  at  C  makes 
with  the  perpendicular  through  P  to  the  radius  vector 
r  =  PC. 

2.  If  a  perpendicular  from  the  pole  P  is  drawn  to  the 
tangent  CF  produced,  and  t  denotes  the  distance  of  its  inter- 
section from  C,  then,  from  equiangular  triangles,  we  shall 


292  LENGTHS  BY   POLAR  CO-ORDINATES. 

bave  ^:PC::FE:CF,    or    t\r\\dr\d2, 

vdi* 
which  gives  dz  =  -—  for  another  expression  of  the  differen- 
t 

tial  in  polar  co-ordinates. 

Otherwise. — Referring  to  the  figure  at  p.  131,  it  is  clear 
that  the  right  triangle  rSN  gives  the  radius  vector 

T-S  =  r  =  SN  sin  N  =  W  sin  N, 
by  using  N'  to  represent  the  normal  SN. 

By  taking  the  differentials  of  the  equation  r  =  W  sin  N, 
supposing  N'  to  be  constant  or  invariable,  we  have 

dr  =  N'  cos  Nc^K  =  N'  cos  N#, 

as  is  manifest  from  what  has  been  shown ;  also,  from  what 
has  been  shown,  we  have  N'o?<^  =  dz,  the  differential  of  the 
arc  AS,  and  of  course  dr  =  cos  'Ndz.  Since  cos  N  =  sin  rSN", 
if  we  multiply  the  members  of  this  by  r,  we  shall  have 
rdr  =  r  sin  rS'^dz,  in  which  ?•  sin  rSN  =  the  perpendicular 
from  r  to  SN,  which  evidently  equals  t. 

Hence,  we  sTiall  have  tds  =  rdr,  or  ds  =  ~-  ,  the  same 

t 

result  as  found  from  the  preceding  method. 

Thus,  to  find  the  length  of  the  logarithmic,  or  equiangular 

7' 

spiral,  since  -  =  the  secant  of  the  angle  at  which  the  radius 

vector  cuts  the  curve,  if  we  represent  the  secant  by  s,  we 
shall  have  dz  =  sdr,  in  which  s  is  constant,  since  the  radius 
vector  always  cuts  the  curve  at  the  same  angle. 

Hence,  by  taking  the  integral,  we  shall  have  j  dz  =  s  j  dr 

or  2  =  sr,  which  needs  no  correction,  supposing  the  integral 
to  commence  with  r ;  and  it  follows  that  z  varies  as  r. 

For  another  example,  we  will  take  the  spiral  of  Archimedes, 
whose  equation  is  r  —  a</). 


LENGTHS   BY   POLAR   CO-ORDINATES.  293 

B  J  taking  the  differentials,  we  have  dr  =  ad^  ;  and  thence 

d2^  j^ {iHf  +  c^/'^)  =  |/(^  +  l)  c//-  =  ^  V ('/•'  +  o?)  dr, 

which  agrees  in  form  with  the  differential  of  the  length  of 
the  common  parabola  given  at  p.  281,  when  we  put  y  and  h 
for  r  and  a.  Hence,  putting  r  and  a  for  y  and  h  at  p.  282, 
we  have 

z  =  ~   ^(.^  +  a^)  +  -log ^ 1 

as  required,  in  which  z  commences  with  r. 

For  further  illustration,  we  will  find  the  length  of  the 
involute  of  the  circle. 

By  proceeding  as  at  pp.  288  and  289,  and  adopting  the 
same  notation  as  there  used,  we  have 

which,  multiplied  by  4/  (r^  —  E-),  taken  for  the  radius  vector, 

gives  dz  =  \/  (/-^  —  R-)  d(f>  =  -^ ; 

whose  integral  gives 

supposing  that  the  integral  commences  with  r  =  B>]  noticing, 
that  in  this  solution  the  pole  is  supposed  to  move  from  A, 
around  the  circumference  of  the  circle,  in  the  order  of  the 
letters  A,  D,  B,  as  at  p.  288. 

For  the  last  example,  we  will  take  the  reciprocal  spiral, 

whose  equation  is  r  =  -  or  6  =z  ~. 
(p  r 

By  taking  the  differentials  we  have 


294  VOLUME  OF  A  SOLID. 

d(l>  —  —,     wliich  gives    d(p'  =  —^-y     or    7'^'d<l>^  =  —^\ 

and  thence  we  have 

dz  =^  ^{fd<t>^  +  d,^)  =  |/(~  +  l^dr=^  ^/{i^''  +  1)  ^ 

rdr  dr 


('^+1)? 


=  dV{7^  +  l)  +  df (log  r)  ^  (^  Slog  [|/(P  +  r^)  +  l]i  ; 
consequently,  by  taking  the  integrals,  we  shall  have 

C  being  the  arbitrary  constant ;  this  integral  is  clearly  the 

same  as  s=  ^ {t^  +  1)  +  log  -_-l^_-  +  C, 

which  will  clearly  enable  us  to  find  the  value  of  z  that  cor- 
responds to  the  interval  between  any  finite  values  of  r. 

(16.)  We  will  now  show  how  to  find  the  contents  or  vol- 
ume of  a  solid,  the  equation  of  whose  surface  can  be  ex- 
pressed by  an  equation  between  the  rectangular  co-ordinates, 
a*,  2/,  and  2,  without  regarding  the  body  as  being  a  solid  of 
revolution. 

It  is  manifest  that  we  may  regard  the  very  small  parallel- 
epiped expressed  by  dxdijdz^  as  being  the  differential  of  the 
solid,  and  represent  its  integral  by  j j  fdxdydz^  by  using  the 

/  successively  to   represent  the  separate  integrations  with 
reference  to  2,  ?/,  and  x. 

Thus,  by  perfbrmiDg  the  first  integration  with  reference  to 
z  taken  between  the  plane  x^  y^  and  the  surface  of  the  body, 
the  integral  is  reduced  to  the  form  /Y  zdxdy ;  which  may  be 


ILLUSTKATED  BY   EXAMPLES.  295 

integrated  again  by  regarding  z  as  being  a  function  of  x 
and  y. 

If  we  at  first  integrate  with  reference  to  y  by  regarding 

X   as   constant,   we    sball   'h.Q.YQ  J  J  zdxdy  =^  I  dx  j  zdy  ^  in 

which  /  zdy  denotes  the  area  of  a  section  of  the  solid  by 
a  plane  that  is  parallel  to  the  plane  of  the  axes  of  z  and  y ; 
then,  having  found  the  integral  /  zdy^  we  can  find  the  in- 
tegral   /  d;c   I  zdy^  which  being  taken  between  proper  limits 

of  0!,  will  give  the  required  volume  of  the  solid. 

It  is  manifest  that  we  may  perform  the  integrations  in  the 

forms  J  J  zdxdy  =  I  dy  I  zdx^  instead  of  using  the  preceding 

forms ;  noticing,  that  those  forms  which  are  the  simplest  in 
the  integrations  are  always  to  be  chosen. 

It  ought  to  be  added,  that  to  find  the  integrals  in  the 
simplest  manner,  the  planes  of  the  co-ordinates  should  be 
drawn,  if  possible,  so  that  they  may  divide  the  body  into 
equal  parts. 

Thus,  to  find  the  volume  of  the  ellipsoid  whose  equation 

f)  O  c> 

is  -^+-4^+-^=l,  it  is  manifest  that  the  planes  of  its 
a-        0'        G^  ^ 

axes  are  those  of  its  co-ordinates. 

Then,  to  simplify  the  equation  still  further,  we  put 
X  =  ax\  y  =  hy',  and  z  =  cz\  which,  by  substitution,  re- 
duce the  equation  to  x'"^  +  y"^  -f  s'^  =  1 ;  the  equation  of  the 
surface  of  a  sphere,  liaving  1  for  its  radius.     Since 

/  zdx  ■—  ac   I  z'dx\      and      /  dy'  I  z'dx'  =  ahcjj  z'dy'dx\ 
it  manifestly  follows,  that  if  we  multiply  the  volume  of  the 


296  VOLUME   OF   A   SOLID. 

sphere  whose  radius  equals  1,  by  the  product  abc,  the  result 
wJU  express  the  volume  of  the  proposed  ellipsoid 

Now,  from  x"'  -f  y'^  +  z''  =  1, 

we  have       s'  =  y  (1  - 1/'  -  x'')  =  ^  {r''  -  x'% 
by  putting  r'^  =  1  —  y'^ ;  consequently,  we  shall  have 

ffz'dy'dx'=ff^/(^r'^  -  x'')dy'dx'= f  dy' f  x/{r''-x'')dx'. 

It  is  manifest  that  to  find  the  integral  /  \/  {r'^  —  x'-)  dx\  r' 

must  be  regarded  as  constant,  and  that  it  will  be  sufiicient  to 
take  the  integral  from  a?'  =^0  to  a?'  =  t\  since  the  whole  in- 
tegral can  thence  be  readily  found. 


It  is  manifest  that 


/ 


^  v'('-"-^'^)&'=^  =  |(i-y"), 


which  equals  the  fourth  part  of  the  area  of  a  circle,  whose 
radius  is  t'  or  y'  (1  —  ^r).     Hence, 

Jdy'  J{^r'^.^x")dx'    becomes     '^f{l-y'')dy', 

whose  integral  it  will  be  sufficient  to  take  from  y'  ^=0  to 

_    /»i  _ 

y'  =  1,  which  gives   |  J  /I  -  v"")  %'  =  g , 

for  the  eighth  part  of  the  sphere,  whose  radius  =  1. 

Hence,  —^ —  is  evidently  equal  to  the  contents  or  volume 

of  the  proposed  ellipsoid,  as  required. 

It  may  be  added,  that,  to  simplify  the  integrals //  zdxdy^ 

we  sometimes  put  y  =  a?w,  and  thence,  since  x  and  y  are 
independent  variables,  get  dy  =  xdu,  which  reduces 

//  zdxdy    to  J  J  zduxdx  z=  I  du  I  zxdx. 


ILLUSTRATED  BY   EXAMPLES.  297 

TTiTis,  in  finding  the  volume  of  the  sphere  whose  equation 
is  a?^  +  2/-  +  2^  :=  E^,  we  have 

2  ^  |/(E2_  x"  -  f)  rr:  ^/(R^  -  x"  -  x'u")  =  ^/[R^  -  x' {1  +  u')], 

by  putting  y  =  xu.     Hence,  the  integrals^^  zduxdx  become 
fdu  f\W  -^  (1  +  %^')'fxdx  =^ 

by  regarding  u  as  constant  in  the  integration,  and  using  C  for 
the  arbitrary  constant.     Supposing  the  integral  to  commence 

when  X  —  0,  we  have  C  —  -^-p. ^  ;  then,  if  the  integral  is 

■p 

extended  to  a?  =  —j- -^  ,  we  shall  have 

|/  (1  +  'i^^) 


=:  —  tan-^^ 


_  _^-    3    ^-"      .« ' 


in  which  the  arc  tan"^  -  must  clearly  be  taken  from  -  =  0 
X   ■  ^  X 

to  -  —  infinity,  and  of  course   the   arc   equals  ^  ;  conse- 

■p3_ 

quently,  ~-^  is  an  eighth  part  of  the  sphere,  and  its  volume 
equals  — ^ —  ,  as  required. 

o 

Remarks. — 1.  If  we  change  the  rectangular  co-ordinates 
X  and  y  into  the  polar  co-ordinates  x  —  7'  cos  <}>  and  y  —  /'  sin  </>, 

13* 


298  POLAR  CO-ORDINATES. 

r  being  the  radius  vector  (in  the  plane  a;,  y^  drawn  from  the 
origin),  and  0  the  angle  it  makes  with  the  axis  of  x ;  then, 
by  assuming  dx  =  —  r  sin  <l>d(j>  from  x^  -\-  if  r=z  r^^  we  have 
ydj  =  rdt\  on  acsoant  of  the  independence  of  x  and  y,  or 

1x1 1'         dr 
dy  =  —  = ;  consequently,  dxdy  =  —  rdrd(f> ;    noti- 
cing, that  if  we  had  assumed  dy  =:  r  cos  <^i</),  we  should, 

dr 

from    ar*  +  y-  =  rK     have  had     dx  = ,     and  thence 

•^  '  cos  0' 

dxdy  =  rdrd(l) ;  so  that  regarding  dr  and  (f0  as  being  posi- 
tive, the  transformations  ought  to  be  taken  absolutely,  or 
without  reference  to  their  signs.  Hence,  zdxdy  will  be 
changed  to  zrdrd<^ ;  which  will  often  be  found  very  useful 
in  integration.  Thus,  to  find  the  volume  of  the  sphere 
whose  equation  is  R-  =  ar -f-  3/-  +  s-  =  r^  +  s-,  we  immediately, 
on  account  of  the  constancy  of  B,  get 

rdr  +  zdz  =  0,     or    rdr  ~  —  zdz, 

which  reduces  zrdrdcp  to  —  Z'dzd({>.     Hence,  we  have 

fj-z\lzd(^  =  -  Irrfz-dz, 

since  the  integml  with  regard  to  ^  ought  clearly  to  be  taken 
throughout  the  whole  circumference.    By  taking  the  integral 

~2n  I  z-dz  from  z  =  'R  to  s  =  0,  we  have 

^      r   ,,         2R«rr 
—  2^  I    zHz  —  — r — 
J  -&  3 

for  the  volume  of  half  the  sphere,  and  of  course  that  of  the 
whole  sphere  is  — — — ;  the  same  as  found  by  the  preceding 

o 

methods. 

2.   It  is  easy  to  perceive  that  we  may  transform  the  infin- 


POLAR   CO-ORDINATES.  299 

itesimal  solid  dzdyilx  by  polar  co-ordinates  after  the  follow- 
in  sr  manner. 

o 

'  Tlius,  let  r  denote  its  distance  from  the  origin  of  the  co- 
ordinates, and  (9  the  angle  it  makes  with  the  plane  of  ir,  y ; 
then,  T  cos  0  being  represented  by  r' ^  it  will  be  the  projection 
of  T  on  the  plane  a?,  y,  and  we  also  have  r  sin  0  =  z. 

Hence,  if  r'  makes  the  angle  </>  with  the  axis  of  x^  we 
shall,  from  what  has  been  previously  shown,  get 

dxdy  =  7''d/d({). 

Since  r^  =  r''  +  s",  if  we  assume  dz  =  r  cos  Odd,  it  results, 
from  the  independence  of  r'  and  z,  that  we  must  assume 
r'd/  =z  rdr ;  consequently,  dzdydx  is  transformed  to 

7^  cos  6drddd(f). 

Hence,  Jjj  dzdydx  =  /  dydxdz^ 

called  a  triple  integral,  is  transformed  to  the  triple  integral 

J   r-  cos  Odrdddcp  —  J  iHr  J  cos  Odd  Jd(f) ; 

noticing,  that  two  successive  integrations  are  called  a  double 
integral,  and  so  on,  according  to  the  number  of  successive 
integrations.  It  may  be  added,  that  the  preceding  trans- 
formation is  essentially  the  same  as  that  of  Laplace,  at  p.  6, 
vol.  II.,  of  the  "Mecanique  Celeste,"  and  that  of  Lacroix,  at 
p.  209,  vol.  II.,  of  his  "Traite  du  Calcal  Integral." 

By  applying  the  preceding  formula  to  find  the  contents  of 
a  sphere  whose  radius  is  R,  it  is  manifest,  as  before,  that  the 
integral  with  regard  to  dd  must  be  taken  through  the  whole 
circumference,  which  reduces  it  to 

J  r-dr  J  cos  OdO  fd(p  =  2n J  r^^dr cos  Odd; 


300  SUKFACES  OF  SOLIDS. 

whose  integral  witli  regard  to  6  must  be  taken  frora 

sin  0  =  —  1     to     sin  0  =  1, 
which  reduces  it  to 

27r  A-W/'  /*cos  Odd  =  infr^dr] 

whose  integral,  with  reference  to  r,  must  be  taken  from 
r  =  0  to  /•  =  R,  which  gives 

47tW 


[Trjr-dr  = 


3    ' 

for  the  volume  of  the  sphere. 

(17.)  We  now  propose  to  show  how  to  find  the  surface  of 
a  body  or  solid,  on  suppositions  like  to  those  in  (16),  and 
shall  premise  the  following  important  proposition : 


Thus,  let  (%c,  oy^  and  oz^  be  three  rectangular  axes  having 
o  for  their  origin  ;  then,  the  square  of  the  numerical  value 
of  the  face  xyz  of  the  triangular  ^pyr amid  oxyz^  equals  the 
sum  of  the  squares  of  the  numerical  values  of  tJie  three  're- 
maining  faces  of  the  jpyramid. 

For  representing  ox^  oy^  and  oz^  severally  by  a,  5,  and  c, 
the  right  triangles  oxy^  oyz^  and  oxz^  severally  give 

V{a'-^l%   V{^'  +  c%   >/(«=  + ^), 
for  the  representatives  of  the  sides  xy^  yz^  and  xz^  of  the  tri- 
angular face  xyz  of  the  pyramid. 

Hence,  since  the  triangular  faces  oxy^  oyz,  and  oxz^  are 


SURFACES  OF  SOLIDS.  801 

severally  represented  by  -^,  -^,  and  -^,  we  propose  to  show- 
that  the  square  of  the  face  xyz  equals 

-4-  +  -4-  "^  X~  4 

If,  for  brevity,  we  represent  the  sides  a?y,  yz^  and  xz^  by 
A,  B,  C ;  by  a  well-known  rule  for  finding  the  area  of  a 
triangle  from  its  three  sides,  we  shall  have  the  area  of  the 
triangle  xyz  expressed  by 

/A  +  B  +  C      B  +  C-A      A  +  C-B      A  +  B-C\* 

( 2—  ^ -1 ■  ^  ^—  ^  2-—)' 

whose  square  equals 

(B  H-  Cy  -  A^      A^  -  (B  -  C)^ 

4        "  ^  "        4 

_  BN-C^-AM-2B  C       A^  -  (B^  +  C^)  +  2BQ 
-"  4  "^  4 

_  2BC  +  (B^+C^-A^)      2BC  -  (B^-  +  C-  -  A^) 
_.  ____        X  -^  ^ 

4B^C^^  -  (B^-  +  C^  -  A^)^ 


16 

From  the  substitutions  of  the  values  y  («^  +  W)^  |/(J^  +  (?\ 
and  ^'{cC-  +  c-),  of  A,  B,  and  C,  in  the  preceding  equation,  we 
have  the  square  of  the  face  xyz  equal  to 

4  (6^  +  (T)  (g^  +  g')  -  4:0'  _  om  +  a^&  -f  l\^ 
16  ~  4  ' 

as  required.  It  is  clear  that  the  triangles  xyo^  yzo^  and  xzo^ 
are  severally  equal  to  the  projections  of  the  triangle  xyz^  by 
perpendiculars  upon  them.  And  since,  from  principles  of 
geometry,  the  perpendicular  from  0  to  the  face  xyz^  multi- 
plied by  it,  equals  the  perpendicular  oz  multiplied  by  the 


302  SURFACES  OF  SOLIDS. 

triangle  yxo^  to  which  it  is  perpendicular,  each  product  being 
three  times  the  pyramid,  it  follows  that  the  triangle  xyo 
equals  the  triangle  xyz  multiplied  by  the  quotient  resulting 
from  the  division  of  the  perpendicular  from  o  by  02^  which 
is  clearly  the  cosine  of  the  inclination  of  the  face  xyz  to  the 
face  xyo. 

Hence,  the  cosine  of  the  inclination  of  xyz  to  either  of 
the  other  faces  multiplied  by  xyz  equals  the  other  face; 
consequently,  from  what  has  been  shown,  it  follows  that  the 
sum  of  the  squares  of  the  cosines  of  the  inclinations  of  the 
face  xyz  to  each  of  the  other  faces  equals  unity  or  1. 

Hence,  also,  any  plane  in  the  plane  xyz  is  such,  that  its 
square  equals  the  sum  of  the  squares  of  its  projections  on 
the  three  planes  xijo^  yzo^  and  xzo. 

We  will  now  suppose  the  curve  surface  to  be  touched  by 
a  plane  at  any  one  of  its  points,  and  that  an  unlimitedly 
small  portion  of  it  at  the  point  of  contact,  having  two  of  its 
opposite  sides  pamllel  to  the  plane  of  ar,  0,  and  the  other  two 
opposite  sides  parallel  to  the  plane  of  y,  z^  is  taken  for  the 
differential  of  the  curve  surface.  Then,  the  projections  of  the 
parallelogram  thus  formed  on  the  planes  a?,  y,  a*,  2,  and  ?/,  2, 
will  evidently  be  parallelograms  whose  areas  may  be  ex- 
pressed by  the  products  dxdy^  dydz^  and  dxdz ;  consequently, 
from  what  has  been  shown,  we  shall  have 

dx-df  +  dy^dz^  +  dxHz"  =  dx'df  fl  +  (^)'  +  (J)'} 

for  the  square  of  the  differential  of  the  curve  surface,  and 
of  course  if  c?-S  represents  the  differential,  we   shall  have 

for  the  required  differential  of  the  curve  BurfBuce. 


SURFACES    OF   SOLIDS.  803 

It  may  be  noticed  that  ~  and  ~,  wliicli  suppose  ^  to  be  a 

(Xi-C  (Xi  'I 

function  of  x  and  y,  have  heretofore  been  represented,  as  in 
Sections  8  and  9,  by^  and  ^;  agreeably  to  which,  if  we  please, 
we  may  write  the  preceding  equation,  according  to  custom,  in 

the  form  (i-S  =  dxd(/\/{i  +jy"+  q^). 

It  may  also  be  noticed,  that  according  to  what  has  been 

shown,     ^(l+y  +  ,=)  =  /{l  +  (J)V(|y 

equals  the  reciprocal  of  the  cosine  of  the  angle  made  by  the 
tangent  plane  with  the  plane  a?,  y. 

To  illustrate  what  has  been  done,  we  will  apply  the  for- 
mula to  find  the  surface  of  a  sphere  whose  equation  is 

^=  -f  r  +  ^'  =  R' 
By  taking  the  partial  differential  coefficients,  we  get 

dz  ^       X        J     (^^  _       y 
dx~~       z  dy  ~^  ~~  z^ 

which  give 

consequentlj,  we  shall  have 

by  putting  R'=  =  R-  —  y\  By  taking  the  integral  relatively 
to  X,  or  by  regarding  E'  as  being  constant,  we  have 

dS  r         dx  -D    •      1   a'       -D    .      1  aj 

Ty  =  V  7(1^^  =  ^  ^'^-'  F  =  ^  ^'°    TW^sO ' 

which,  taken  from  »  =  0  to  a?  =  V(^^—  f)^  gives  ^  =  ^5  j 


804  ARBITRARY   CONSTANTS  ILLUSTRATED. 

consequently,  dS  =  -^-  dy^  whose  integral  is  S  =  -y  y,  which, 

■pi>_ 

taken  from  y  =  0  to  ?/  =  R,  is  -^- ,  the  eighth  part  of  the 

surface  of  the  whole  sphere,  which,  of  course,  equals  4:R-7r, 
four  times  the  area  of  a  great  circle  of  the  sphere. 

Otherwise. — By  putting  the  equation  of  the  spheric  surface 
in  the  form 

R2  =  ar*  +  /-  -f  s'  =  r'  +  2', 
we  shall,  by  the  notation  at  p.  298,  get 

dxdy  =  rdrd(p  =  —  zdzdtp^ 

and  thence    <^S  =  —  zd2d(()  x  -  =  —  d^Rdcj) ; 

whose  integral  relatively  to  (f>  must  clearly  be  taken  through- 
out the  entire  circumference,  and  gives  ds  =  —  2B,7Td2 ;  and 
the  integral  of  this  must  evidently  be  taken  from  3  =  —  R 
to  s  =  R,  which  gives  ^ttR^  for  the  whole  surface  of  the 
sphere,  the  same  result  as  by  the  preceding  method. 

(18.)  We  will  now  proceed  to  show  the  use  of  arbitrary 
constants  in  the  development  of  functions,  and  in  the  integra- 
tion of  differential  equations,  move  than  has  yet  been  done. 

1.  To  show  the  use  of  constants  in  the  development  of 
functions,  we  will  give  the  following  investigation  of  Taylor's 
Theorem. 

Thus,  suppose  the  differential  of  any  function  of  a?  +  A 
may  be  represented  by  the  form 

dF{x-\-h)  =  'F'{x  +  h)dh, 

when  the  differential  is  taken  on  the  supposition  that  A  alone 
is  variable.  By  taking  the  integrals  of  the  members  of  the 
equation,  we  have 


ARBITRARY  CON"STAN"TS   ILLUSTRATED.  305 

F  {x  +  h)  =  C  +fY  {x  +  h)  dh', 
in  wliicli  F  (a?  +  h)  is  tlie  integral  of  the  exact  differential 
f/F  (;»  -f  A),  and  C  tlie  arbitrary  constant,  while  /  F'  (a?  +  A)  dh 

indicates  that  the  integral  of  F^  {x  +  h)  dh  is  to  be  found,  on 
the  supposition  that  h  alone  is  regarded  as  variable.     If  we 

determine  C  on  the  hypothesis  that  the  integral  /  F'  {x  +  h)  dh 

vanishes  when  h  equals  naught,  since  A  =  0  reduces  F  {x+h) 

to   F  (a?),   and  /  F'  (a?  +  A)  dh  to  naught,   we    shall  have 

F  {x)  =  C.     Hence,  by  substituting  this  value  of  C,  the 

equation  F  (aj  +  7^)  =  C  +   I  ¥'  {x  +  h)  dh 

is  reduced  to  F  (a?  +  h)  —  F  {x)  +  J  Y'  {x  +  h)  dh ; 

noticing,  that  F  (a?)  is  not  supposed  to  be  unlimitedly  great 
Because  F'  (a?  +  h)  is  a  function  of  x  +  A,  it  follows,  from 
what  has  been  done,  that  for  F'  (a?  +  h)  we  may  put 

Y{x)+f'F"{x  +  h)dh, 
which  reduces  the  preceding  equation  to 

F  (a?  +  A)  ==  F  (x)  +  Jy  (^)  dh  +  fdhfw  {x  +  h)  dh 
:=¥{x)-{-W{x)h+f'¥''{x  +  h)dh'', 
since  /  F^  (x)  dh  becomes  F'  {x)  h  on  account  of  the  con- 
stancy of   F^(a?),  and  by  using  /     (according  to  custom) 
iQxJJ .     Similarly,  because  F^^  {x  -f-  A)  may  be  represented 
by  F"  {x)  +  Jy"  {x  +  A)  dh,  we  have 


806  ARBITRARY   CONSTANTS  ILLUSTRATED. 

pT'  {X  +  h)  dh'  =fdhfw'  {x)  dh  +  f  Y"  {x  +  h)  dh\ 
and  hence 

F(^  +  A)  =  F(a.)  +  r{x)j  +  Y'\x)  ^  -\-fY%c  +  h)dIi\ 

If  n  represents  any  positive  integer,  it  is  manifest  that  we 
shall  in  this  way  get 

To  find  the  values  of  F'(i»),  V  {x\  Y"  {x),  kQ.,wQ  resume 
the  proposed  equation 

d¥{x  +  h)  =  l^'{x  +  h)d/i, 

I,-  -u    •                 d¥{x-\-h)      T^,,     ,    ,. 
which  gives  — -~. =  ¥' [x  +  A), 

for  which  we  may  evidently  put 

^^--tA)  =  F  (aj  +  A)  =  F'  {x)  +fY'  {x  +  h)  dh; 

for,  since  x  and  A  enter  the  function  F  (a?  -}-  A)  in  the  same 
manner,  it  is  clear  that  the  differential  coefficient  taken  by 
regarding  h  alone  as  variable,  must  be  equal  to  its  differen- 
tial coefficient,  taken  by  regarding  x  alone  as  variable. 

Because    F'  {x)  enters  the  preceding  equation,   like  the 
arbitrary  constant  C  in  the  equation 

F  {x  Jrh)=Q+  Jy  {x  +  h)  dh, 

it  is  manifest  that  we  may  determine  F'  (.^)  from  the  equation 

&J^±]!l^^'^,)+fr'i.  +  k)dh, 


ABBITRARY   CON'STANTS   ILLUSTRATED.  307 

on  the  supposition  tliat  when  h  =  0,  we  must  also  have 

fF'Xx  +  h)dk  =  0; 
consequently,  by  putting  A  =  0,  we  get 

Because  the  equation 

F'(»  +  A)  =  F'(^)  +f¥"{x  -f  h)  dh 
may  be  supposed  to  have  been  obtained  from 
dY{x  +  h)  =  Y'{,iG-^h)dh, 
in  the  same  way  that 

F  (a?  +  A)  =:  F  ix)  i-fr  {x  +  A)  dh 
has  been  derived  from 

^F  (^  +  A)  =  F'  (,^  -f  A)  dh, 
it  is  clear  that  we  shall  (as  before)  get 

flY  (x)     ,         . 
Because  F^  (x)  =  — ---^ ,  if  dx  is  constant,  it  is  ciear  that 

.     -,,,,.       d.W{x)  .     d'¥{x)    .       ,.  ,   d'F{x) 

ior  J^    {x)  =  — -—^  we  may  write  — •,-y-^ ;  m  wnich      ■■  j  ■ 
(XX  ci  x"  (Xxr 

is  called  the  second  differential  coefficient  of  F  {x).     It  is  evi- 
dent that  we  shall  in  like  manner  get 

and  so  on,  for  the  third,  fourth,  &3.,  diiferontial  coefficients. 
Hence,  we  shall  have 

I  (^  +  A)  _  F (..)  +  ^^.  ^  +  --j^  ^^  +  ^-^-  j~2j  +, 

i&c,  as  in  Taylor's  Theorem,  as  required. 


808  ARBITRARY  CONSTANTS  ILLUSTRATED. 

It  will  be  perceived  that,  in  the  precediug  investigation, 
we  have  virtually  introduced  an  unlimitedly  great  number 
of  constants ;  since  there  must  (essentially)  be  as  many  as 
there  are  equations  like 

F  {x-{-h)  =  C  -hf¥'{x+h)  d/i  =  ¥  {x)  +fF'  {x-\-h)  dh, 

Y{x^-K)  =  F'  {x)  ■\-  Jf"  (^  +  h)  dh,  and  so  oa 

But  since  these  constants  all  result  from  C  =  F  (a?),  or  are 
dependent  on  C,  it  is  clear  that  the  integral  of 

d'F{x  +  h)  =  Y{x-\-h)dh 

contains  only  one  arbitrary  constant.  Indeed,  it  is  manifest 
that  in 

enter  as  constants ;  whose  values  result  from  </>  (x),  or  depend 
on  X  and  the  form  of  the  function  represented  by  0. 

It  is  hence  evident,  that  in  integrating  any  differential 
equation  there  will  be  as  many  constants  introduced  as  there 
are  integrations,  which  will  be  arbitrary  when  they  are  inde- 
pendent of  each  other. 

2.  Supposing  an  equation  between  variables  and  constants 
to  be  freed  from  fractions  and  radicals,  and  that  its  terms  are 
all  brought  into  the  first  member  of  the  equation,  and  ordered 
accordmg  to  the  ascending  or  descending  powers  of  one  or 
more  of  the  unknown  letters,  then,  if  the  equation  has  a  term 
called  the  absolitte  term^  which  does  not  contain  any  variable, 
by  taking  the  differential  of  the  equation,  the  absolute  term 
will  disappear  from  the  differcDtial  equatic^ii;    and  the  pro- 


ABBITRAUY   CONSTANTS   ILLUSTRATED.  809 

posed  equation,  sometimes  called  the  ^?rz7/247?'t'^,  is  said  to 
have  lost  a  constant  in  the  ditferential  equation,  sometimes 
called  the  first  derivative  of  the  proposed  equation,  bj  a 
direct  differentiation  of  the  primitive ;  but  if  the  form  of  the 
primitive  is  changed,  so  as  to  make  the  constant  coefficient  of 
any  other  term  of  the  equation  the  absolute  term  of  the 
changed  equation,  its  absolute  term  will,  as  before,  disappear 
from  its  differential  equation,  which  may  be  called  an  indi- 
rect derivative  of  the  proposed  equation,  which  may  be  said 
to  have  resulted  from  an  indirect  differentiation  of  the  pro- 
posed equation.  It  is  hence  easy  to  perceive  that  there  may 
be  as  many  direct  and  indirect  differential  equations  obtained 
from  the  given  primitive,  to  free  it  from  each  of  its  constants 
separately,  as  it  contains  constants. 

Thus,  if  y  -f-  «a3  +  ^  =  0  represents  the  given  equation,  hav- 
ing h  for  its  absolute  term,  then,  by  a  direct  differentiation  of 

the  equation,  we  get  dy  +  adx  =  0  or  -~-  -\-  a  =  0  for  the 

direct  derivative  of  the  proposed  primitive,  which  does  not 
contain  the  absolute  term  b.     By  putting  the  proposed  equa» 

tion  under  the  form a  =  0,  we  have  a  for  its  abso- 

X  ' 

lute  term ;  then,  taking  the  differentials  of  the  members  of 
this,  we  have 

y_+h  _  d{y  +  h)  x  x  —  dx (y  +  h)  _ 
X  m?  .       ~~    ' 

or  xdy  —  ydx  —  hdx  =  0, 

or  its  equivalent  y -~-  +5  =  0, 

which  is  the  indirect  derivative  of  the  proposed  equation, 
which  is  clearly  the  same  result  that  the  elimination  of  a 
from 


810  ARBITRARY   CONSTANTS  ILLUSTRATED. 

y  +  ax  -\-h  =  0    by     -^  -\-a  =  0 

will  give ;  it  is  also  clear  that  the  elimination  of  -j-  from  the 
differential  equations 

will  reproduce  the  proposed  primitive.  It  is  also  manifest 
that  the  derivative  equations 

-^  +  a=0    and    y—  -j^  -f  ^  =  0 
dx  ^        dx 

are  entirely  distinct  from  each  other ;  the  equivalent  of  the 

first  dy  +  adx  =  0 

being  immediately  integrable,  while  the  integral  of  the 
equivalent  of  the  second 

ydx  —  xyd  -\-hdx  =  0     (or    —^--j^ 2    =  0) 

8/  X 

becomes  integrable  after  it  is  multiplied  by  —  -^ ,  the  factor 

which  is  said  to  he  requisite  to  the  integrahility  of  the  in- 
direct derivative,  ydx  —  xdy  +  hdx  —  0,  of  the  proposed 
primitive. 

If  we  take  the  equation  y  -^1)X  ■\-  car  ^=  0,  it  is  evident 
that  a  constant  can  not  be  eliminated  from  it  by  a  single 
direct  differentiation,  while  the  constants  h  and  c  can  be 
eliminated  by  indirect  differentiations.  For,  by  putting  the 
equation  under  the  forms 

-,  H h  C  =  0    and    ^  -r  cx-^h  =  0, 

mr       X  X  ■ 

and  taking  the  differentials,  we  have 


p 


REDUCTION  OF   INTEGRALS   TO   SIMPLER   FOEM&       811 

d%+d-=0    or    ^*' _  (2y  +  fo)  =  0, 

X'  X  ax 

and  cZ  -  +  cdx  —  0     or    x  ^  —  (y  —  car)  =  0. 

X  dx       ^  ' 

It  is  evident  that  by  eliminating  -~  from  these  equations,  we 

dx 

shall  get  the  primitive  equation  y  -\-  hx  -\-  cx' ^^  0,  which 

can  not  be  found  from  the  immediate  integration  of  either  of 

the  derived  equations. 

If,  for  another  example,  we  take  the  equation 

.    y  —  aa?  +  a^  ==  0 ; 

dxi 

then,  by  differentiation,  we  have  dy  —  adx  =  0  or  -~  =a. 

Substituting  -~  for  a,  in  the  proposed  equation,  it  becomes 

xdy       df 
y        dx   ^  dx'       ^' 

which  is  of  the  second  degree  in  ~  ,  and  of  the  iSrst  order  of 

differentials.  Thus  we  perceive  how  differential  coefficients 
of  the  higher  orders  may  sometimes  be  introduced  into  differ- 
ential equations,  by  eliminating  the  different  powers  of  a 
constant  from  it,  by  means  of  the  powers  of  a  differential 
coefficient;  but  it  is  manifest  from  the  methods  of  finding 
multiple  points  in  Section  YII.,  that  they  may  sometimes  be 
introduced  by  differentiating  as  in  finding  multiple  points. 
(See  the  examples  at  p.  191,  &c.) 

3.  We  now  propose  to  show  how  to  reduce  such  integrals 

as  are  of  the  forms  /    X.dx"\  j   Xdx"",  &c.,  vi  and  ?i  being 

positive  integers,  to  simple  integrals^  expressed  by  the  sign  /  . 
Thus, 


312      REDUCTION   OF   INTEGRALS   TO  SIMPLER   FORMa 

f\da?  =zfdxfxdx  =  fdxf{Xdx  H-  x-yidx  -  X^rda?) 

=  X  I  X^dx  —  I  'Kxdx  ; 

which  clearly  results  from  integrating  by  parts  (see  p.  260). 
Similarly, 

f'xda^  =fdxf'xda^  =f{xdxfxdx  -  dxfxxdx) 

=  ^  {a^fxdx  -  2xfxxdx  +  fxx'dv), 
fxdx'  =Jdxfxd^ 

=  ^  fix'fxdx  -  2xdxfxxdx  4-  dxfx3?dx) 
=  j^  [x'fxdx  -  Sx'f  Xxdx  +  Zxfx^dx-fX;^dr), 
and  so  on,  to 

A<*^"  =  i.2.3..'(,>-r)  i^-'f^^-  -  ^  ^-i^-^ 

whose  law  of  continuation  is  manifest    (See  Lacroix,  voL  II., 
p.  152.)     If  for 

/  Xdx^   I  Xxdxy  I  Xx^dx^  &c., 

in  the  preceding  formula,  we  put 

fxdx  +  cfxxdx  +  C\fx/dx  +  C",  &c., 

in  which  C,  C,  C",  &c.,  are  the  arbitrary  constants,  they  will 

represent  the  complete  integrals  indicated  by  /   Xdx" ;  be- 
cause there  will  be  as  many  arbitrary  constants  as  there  are 


LIMITS  TO   INTEGRALS.  813 

integrations,  and  tliey  clearly  enter  the  formula,  as  they 
ought  to  do. 

If  the  constants  equal  naught,  it  is  clear  that  the  pre- 
ceding formula  is  equivalent  to 

provided  y  is  regarded  as  independent  of  x  in  the  integration, 
and  that  the  integral  is  taken  from  the  value  of  x  at  the 
commencement  of  the  integral,  to  the  value  of  x  at  the  end 
of  it ;  for  which  last  value  (of  x)  we  ought  to  put  y,  or  y 
must  represent  it 

Eemarks. — 1.  The  preceding  formula  enables  us  to  find 
limits  to  the  integrals  indicated  by 

given  in  the  investigation  of  Taylor's  Theorem,  at  p.  306. 

For  X  may  represent —j^ — -^  and  h  may  be  used  for 

X  in  the  preceding  formula ;  consequently,  we  shall  have 

If  we  put  y  —  h  =  yz^  or  h  =  y  (I  —  z)^  we  shall  have 
dh  ~  —  ydz^  since  y  is  independent  of  h ;  consequently,  we 
shall  get 

r^'^' = - 1.2.8  ■■'■(«-i)  /%"^"-'^-. 

supposing  the  integral  to  be  taken  from  3  =  1  or  A  =  0  to 
z  —  0  or  y  =  h.  If  the  limits  of  the  integral  are  interchanged, 
it  is  evident  that  we  shall  have 

rXd/i^^=-— -. ~  fxy"z--'d2.    . 

v  1.2.3 {n  —  1)  J      -^ 

If  M  and  m  are  the  greatest  and  least  values  of  X  (re- 
14 


814  REPRESENTATIONS   OF   INTEGRALS. 

garded  as  having  the  same  sign  and  as  finite),  in  the  interval 
trom  a?  to  ic  -f  A,  then  we  shall  have 

A'^^"-i.2....;i-i)A^"-"-''^-' 

such  that  T-K-7Z and  ^  ^  ^— are  its  greater  and 

1.2.3. ...  71  1.2.3 n 

less  limits ;  noticing,  that  these  limits  are  clearly  the  limits 
of  the  errors  committed  by  rejecting 

1'^''"=  1.2.3.  ■■'(.-dA^"^"-'^- 
(See  Lacroix,  vol.  Ill,  p.  398.) 

2.  It  is  easy  to  find  the  integrals  indicated  by   /  Xc?a?",  in 

such  a  way  that  they  shall  be  freed  from   /  ,  the  sign  of 
integi-ation.     Thus,  since 

r.   (^         dX  a^        (^X    ar"  ,     \ 

(see  Bernouilli's  series  at  p.  261),  and  by  disregarding  the 
arbitrary  constants  (for  the  present),  we  shaU,  by  integrating 

by  pai-ts,  get  /  Xdx^  = 

"    1.2        dx  1.2.3  "^  da^  1.2.3.4        dx'  1.2.3.4.5  '^' 
From  this  result,  we,  in  like  manner,  get 

which,  integrated  by  parts,  as  before,  gives 
J   Ar/x_X— --  —  ----  +  — j^^—-,&c. 


KEPRESENTATIONS   OF   INTEGRALS.  815 

Proceeding  in  this  way,  and  supplying  the  arbitrary  con- 
stants, it  is  easy  to  perceive  that  we  shall  have 


/ 


Xdx-  =  X 


1.2 n       dx  1.2 {fi  +  1) 

nOi_+l)^t!  n  (71  +  1)01  +  2)  3 

d'X  1.2  d'X~        1.2.3 

"^  dx"   1.2...,  {71+  2)       dx'        1.2....  {n  +  S)        "^ 

C,  C\  &c.,  being  the  arbitrary  constanta     (See  Lacroix,  vol. 
IL,  pp.  154  and  155.) 

Being  now  prepared,  we  will  give  a  short  section  on  the 
Calculus  of  Yariations. 


SECTION  II. 

FIRST  PRINCIPLES  OF  THE  CALCULUS  OF  VARIATIONS. 

(1.)  If  y  is  an  arbitrary  variable,  whicb  depends  on  a  con- 
stant ;  then,  if  in  consequence  of  a  change  in  the  constant  it 
becomes  Y',  the  difference  Y'  —  V,  represented  by  (JY,  is 
called  the  variation  of  Y,  which  is  expressed  by  writing  (J, 
called  the  characteristic  of  variations^  before  or  to  the  left 
of  Y.  If  0  (Y)  represents  any  function  of  Y,  and  the  alge- 
braic sum  of  all  the  changes  in  the  value  of  <t>  (Y)  that  result 
from  the  separate  variation  Y'— Y,  represented  by  dY,  of  each 
Y  in  0  (Y)  is  taken,  it  will  represent  what  is  called  the  variation 
of  <p  (Y) ;  which,  as  before,  is  expressed  by  writing  the  char- 
acteristic <J  before  or  to  the  left  of  the  function;  so  that 
6<p  (Y)  stands  for  the  variation  of  the  function  0  (Y). 

(2.)  From  a  comparison  of  the  preceding  definitions  with 
those  of  a  differential  of  a  variable  and  a  function  of  it  [see 
(4)  at  p.  2],  it  is  easy  to  perceive  that  we  shall  have 


— -tW-  being  the  differential  coefficient,  regarding  Y  as  being 

the  independent  variable.     Hence  we  shall  have 

6(t>  (Y)  _  dct>  (Y) 
6Y     ^     dY    ' 

which  shows  that  the  variational  and  differential  coefficients 


CALCULUS   OF   VARIATIONS.  817 

of  a  function^  loiih  reference  to  the  same  variable^  are  equal 
to  each  other. 

(3.)  Since  from  (1.)  Y' =  Y  -\-  dY,  we  have,  from  Taylor's 
Theorem, 

•    ,^(V')  =  0(V  +  <5V)  =  0(V)  +  ^-^  <5V  +,  &o.  ; 

which,  bj  retaining  only  the  term  that  contains  the  simple 
power  of  cJYj  becomes 

which  clearly  shows  that  0  (Y')  must  be  of  a  different  form 
from  (/)  (Y),  since  6Y  results  from  the  change  of  a  constant 
contained  in  Y. 

Hence,  if  we  represent  the  proper  form  of  the  first  mem- 
ber of  the  equation  by  V  (Y^),  we  shall  get 

V(VO  =  0(Y)  +  ^cyY; 

which  gives         i/,(YO-0(Y)  =  ^^(5Y. 

Since       J^    6Y  is,  according  to  what  has  been  shown,  equal 

to  (5  0  (Y),  we  shall  hence  get 

^(Y')-cp{Y)^6<p{Y). 
By  taking  the  differentials  of  the  members  of  this  equation, 
we  have  dxj)  (J')  —  d(f>(J)  =  d6(f>  (Y) ; 

or  since  d'lp  {Y')  is  a  change  of  the  fonn  d(p  (Y),  we  shall 
have  dijj  (YO  -  di>  (Y)  =  ddcp  (Y), 

and  thence  d6(f>  (Y)  =  ddcp  (Y) ; 

and  with  equal  facility  we  get 

C^M0(Y)  =  <5ri'^0(Y), 


818 


CALCULUS  OF  VAIIIATIOKS, 


n  being  a  positive  integer.     Hence,  m 

or  in  any  expression  to  which  d""  and  6  are  prefixed,  we 
may  clearly  interchange  d  and  (5,  the  characteristics  of  differ- 
entials and  variations,  without  affecting  the  value  of  the 
result ;  noticing,  that  this  is  usually  considered  as  being  the 
fundamental  principle  of  the  Calculus  of  Variations. 

On  account  of  the  importance  of  what  has  been  done,  in 
what  is  to  follow,  we  propose  to  illustrate  it  geometrically. 


Thus,  if  the  line  OC  is  taken  for  the  line  of  the  abscissas, 
on  which  the  positive  values  of  V  are  estimated  from  the 
origin  0,  toward  the  right;  then  ab,  being  drawn  as  an 
ordinate  to  the  curve  he,  representing  the  value  of  0(V), 
which  corresponds  to  Oa  =  V,  by  changing  Oa  or  V  into 
OA  or  V,  and  drawing  AB  parallel  to  ab  to  represent  the 
changed  value  of  ab=z(f)(Y)  as  an  ordinate  V^(V"),  in  the 
changed  curve  BD,  we  shall  have  i/^  (V)  =  ^  (V),  the  varia- 
tion of  ab  represented  by  AB  —  ab  in  the  figure. 

Similarly,  Oc  and  OC  representing  other  values  of  V  and 
Y^,  we  shall  have  CD  —  cd  for  the  representative  of  the  cor- 
responding value  of  V^  {Y')  —  </>  (Y),  which  may  be  regarded 
as  consecutive  to  the  preceding  value. 


CALCULUS   OF   VARIATIONS.  819 

Hence,  we  sliall  have 
(CD  -  cd)  -  (AB  -  al)  =  (CD  -  AB)  -  {cd  -  ab) ; 

the  first  member  of  this  equation,  from  the  definitions  at 
page  2,  being  the  differential  of 

( AB  -  ah)  =  W  -cpY^Scp  (V), 

since  (AB  —  ah)  is  on  the  same  curves  with  its  consecutive 
value  (CD  —  cd) ;  while  {cd  —  ah)  in  the  second  member  of 
the  equation  has  (CD  —  AB)  for  its  consecutive  value,  which 
is  taken  in  the  curve  BD  and  not  in  the  curve  he ;  and  of 
course,  since  (cd  —  ah)  =  d^  (Y),  we  shall  have 

(CD  -  AB)  -  (cd  -  ab) 

expressed  by  ^d(l)(Y).  Hence,  from  what  has  been  done, 
we  shall  have  ddcf)  (Y)  =  ^d(f>  (V) ;  which  agrees  with  what 
has  been  shown,  from  other  considerations. 

Again,  since  Oa  =  Y,  and  ac  =  dV,  and  cC  =  cJ  (Y  +  <JV), 
we  shall  have      OC  =  Y+  dY+  d(Y+  dY) ; 
also,  from  Oa  =  Y,    and   a  A  =  cJY, 

together  with      AC  =  d(Y  -{-  (5Y), 
we  have  QC  =  Y  +  (5Y  +  d(Y  +  ^Y). 

Hence,  from  equating  these  values  of  OC,  we  have 
Y+  dY  +  6(Y+  dY)  =  Y+  (5Y+  d(Y+  dY), 
which  is  easily  reduced  to  6dY  =  d'6Y. 

If  AB  and  CD  coincide,  in  direction,  with  ah  and  cd,  or 
if  A  falls  on  a  and  0  on  c,  it  is  clear  that  the  equation 
SdY  =  d6Y  will  not  exist. 

(4.)  There  is  an  analogous  principle,  with  reference  to  the 
signs  of  integration  and  variation,  which  we  will  now  pro- 
ceed to  notice. 


320  CALCULUS  OF  VARIATION'S. 

Thus,  if  we  put  the  integral  indicated  bj  /  u  equal  to  Y, 

bj  taking  the  differential  we  shall  have  u  =  dY ;   whose 
variation  gives 

6u  =  6dV  =  (bj  interchanging  6  and  d)  ddY, 
whose  integral  gives 

fdu  =  6Y  =  dfu. 

In  like  manner,  b  j  representing  the  nth  integral    /   uhyY, 

a*nd  taking  the  wtb  differentials  of  these  equals,  we  shall 
have  u  =  d^'Y ;  whose  variation  is 

6u  =  Sd^'dY  =  d'^dY, 

whose  nth  integral  gives 

/n  /*n 

6u=:dY  =  6j    U. 

Hence,  if  the  characters  /  and  6  are  prefixed  to  anj  ex- 
pression, they  may  he  interchanged  without  affecting  its  value, 

(5.)  If  t  in  any  calculation  represents  the  independent  varia- 
ble, then,  since  dSt  =  ddt,  and  that  dt  is  constant,  we  shall  have 
ddt  =  0,  since  dt  is  invariable ;  consequently,  from 

dSt  z=6dt  =  0    we  have     dSt  =  0, 

whose  integral  is  6t  =  const 

Hence,  the  variation  of  the  independent  variable  is  con- 
stant^ or  invariable. 

(6.)  To  illustrate  what  has  been  done,  and  to  show  the 
nature  of  variations  more  fully,  we  will  take  the  following 


EXAMPLES. 


1.  To  find  tbe  variations  of  y«,  sp'^,  a?y,  and  -. 


EXAMPLES.  821 

By  differentiating  and  using  6  for  d  (see  p.  316),  we  readily 
get 

--yn     dy^-'^-x    1      ^x,   7j6x^x6y,    and    -^ , 

for  the  variations. 

iicLx  i/(i.u 

2.  Given  t  =  ~~  and  6^  —  ^-  ,  to  find  tlie  variations  of 

ay  ax 

t  and  5,  when  dy  is  regarded  as  constant. 
Differentiating,  and  using  6  for  d^  we  get 

_  Sydx  +  yddx  _  dydx  +  yddx 
~  dy  ~  dy  ' 

T  _  Sydydx  —  ydydSx 

~  dx^ 

3.  Given  du  =  mdx  +  7n'd'^x  +  ndy  +  n'd^y  +  pd\  to 
find  the  variation  of  u. 

This  is  clearly  effected  by  changing  either  d  in  the  several 
terms  into  (J,  which  gives 

6ii  z=i  mdx  +  iii'd^x  +  ii^y  +  n'ddy  +  pdMz  = 
(agreeably  to  what  has  been  done) 

tndx  +  tn'ddx  +  ?i(5y  +  n'ddy  +  jjd-dz^ 
as  required.     This  is  the  same  as  to  change  the  last  d  into  6, 

4.  To  free  the  variations  under  the  sign  /  ,  in  the  integrals 

/  pddx  —  J  pdSx,     and     /  qddy  =  I  qd^dy, 

from  the  sign  c?,  of  differentiation. 

Integrating  by  parts,  these  expressions  are  readily  reduced 

to  /  pddx  =  pSx  —  I  dp6x, 

and    /  qd^Sy  =  qdSy  —J  dqddy  —  qddy  —  dqdy  +  /  d^qSy^ 
as  required. 


322  CALCULUS  OF  VARIATIONS. 

5.  To  find   the   variation   of  the   integral   /  VdJ^-\-  dif^ 

which  indicates  the  length,  or  rectification,  of  an  indefinite 
arc  of  a  plane  curve,  or  line,  when  referred  to  rectangular 
co-ordinates ;  noticing,  that  such  an  integral  is  sometimes 
called  an  indefinite  or  unlimited  integral, 

Bj  taking  the  variation,  regarding  both  dx  and  dy  as 
variable,  and  interchanging  the  signs  of  integration  and 
variation,  we  have 

df  Vdur  +  dy'  =:f6  Vdi?  +  dy" 

=  f(  _^___  6d.  ^    -JM=  6dy) 
•^  ^  Vdx"  +  dy'       .       Vdx'  +  dy'      '^/ 

(by  interchanging  6  and  d,  and  integrating  by  parts) 

=  r-J^^  ds,  +  r—M^dSy 

J    Vd.i^  +  dy"  J    Vdx'  +  dy' 


Vdx'  +  dy'  Vds(^  +  dy' 

d^      A.     r^      ^^y 


/d 6x  —    /  I 
Vdx'  +  du'  J 


6y. 


Vdx'  +  dy'  J       Vdx'  +  dy' 

It  will  be  perceived  that  the  preceding  integral  consists  of 
two  sorts  of  terms :  one  of  which,  that  clearly  relates  to  the 

limits  of  the  integral,  is  freed  from  the  sign  /  ;  while  the 
other  terms  are  under  /  ,  or  their  integrals  are  to  be  taken. 

If  a?',  y\  and  x'\  y" ^  are  the  co-ordinates  of  the  first  and 
last  extremities  of  the  integral,  then,  the  integral  taken  be- 
tween the  preceding  limits,  becomes 


EXAMPLES.  823 

/x".  y"    , dx'^  -  ,,  dy"  ^  ,, 

\Vdx'^  +  dy'^  Vdx''  +  dy"'       / 

^    rd-^^=-_6x-fd^M=6y, 
J      Vdx-  +  dy'  ^      Vdx^  +  dy^ 

If  tlie  extremities  of  the  integral  are  fixed  points ;  then, 
it  is  evident  that  6x"^  Sy'\  ^x'^  6y\  will  each  equal  naught, 
and  the  integral  will  be  reduced  to 


Vdx^+dy' 


•  fd  -^=.  6X  -   fd-^—,y 

^       Vdx'  +  dy^  -^      Vdx'^-^-dy^ 

If  the  extremities  of  the  integral  are  always  on  given  lines, 
then  ^x'\  dy"^  will  be  connected  by  the  equation  of  the  line 
at  the  end  of  the  integral,  while  6x\  6y\  will  be  connected  by 
the  equation  of  the  line  at  its  first  extremity. 

If  the  integral  is  to  be  exact  or  freed  from  the  sign  /  ,  so 

as  to   leave  Sx  and  ^y  arbitrary,   then,  it  is  clear  that  we 
must  have  the  separate  equations 

fd—J^^6x^0    and      f d —.M=.-=.  dy  =  0, 
^       Vdx'^  +  dy'  ^       Vdx'  +  dif 

«md  bf  cause  dx  and  6y  must  be  arbitrary,  these  equations 
inus^  1;e  satisfied  by  assuming 

-         ^^    _  =  0     and     d^M=L=  =  0, 


Vdx""  +  c/y-  Vdx'  +  dy" 

who',0  integrals  give 

— ; ^' =  const,  and     — :=— =  const. 

Vdx"  +  dy^  Vdx"  +  dy^ 


324:  CALCULUS  OF  YARIATIONa 


consequently,  eliminating  VcLir  +  dt/  from  tliese  equations, 

we  have  f-  z=  a  =  const.,     or    dy  =  adx^ 

whose  integral  gives  y  =  ax  +  by  the  equation  of  a  straight 

line,  whose  constants  are  a  and  h.     Hence,  the  variation  of 

the  proposed  integral  being  exact  and  its  limits  fixed  points, 

it  is  evidently  reduced  to  naught,  or 

.c".  y"    , 

Vdz^  -j-  c?/  =  0. 


•/: 


It  is  also  manifest,  that  the  straight  line  represented  by 

y  =  ax  +  h,     makes      /*^  '  ^   Vdx"  +  dy% 

a  minimum ;  such,  that  its  value  can  easily  be  found  from 
the  co-ordinates  a?',  y\  and  x^\  y'\  of  the  fixed  points  at  the 
extremities  of  the  integral.  For  by  putting  x'  and  y'  in 
yz=zax-\-l)  we  have  y'  =  ax'  +  h  and  by  putting  x''  and  y" 
for  X  and  y  in  y  =  aa?  +  J  we  also  have  y"  =  ax''  -{-  h ;  con- 
sequently, the  solution  of  the  equations 

y'  =  ax'  -\-h    and    y'^  =  ax"  +  5, 

will  give  the  values  of  a  and  5,  and  thence  the  required 
straight  line  can  be  drawn. 

Vd3(^  +  dy^ 

to  be  exact,  and  to  be  put  equal  to  naught,  we  shall,  as  be- 
fore, have  y  =  ax-{-h,  the  equation  of  the  straight  line, 
together  with  the  .equation 


Vdx"^  +  dy"^  Vdx'"  +  dy' 


EXAMPLES.  825 

which  results  from  the  variations  at  the  limits  of  the  pro- 
posed integral. 

If  the  limiting  curves  at  the  extremities  of  the  integi-al 
are  independent  of  each  other,  then  it  is  clear  that  the  pre- 
ceding equation  reduces  to  the  two  separate  equations 

dx"6x''  +  dy"6y"  =  0     and     dx'dx'  +  dy'dj/'  =  0, 
or  their  equivalents 

■     ^'Jl  +  I-O    and    ^l^'^  +  l-O- 

noticing,  that  dx^\  6y"^  and  6x\  6y\  are  supposed  to  belong  to 
the  limiting  lines,  while  dx^^^  dy'\  and  dx' ^  dy' ^  belong  to  the 
extremities  of  the  straight  line  y  =  aa?  +  5.     Hence,  from  the 

equation  of  the  straight  line  we  have  —-,  ==  a,  where  it  meets 

the  first  limiting  line,  which  reduces 

^y'^  +  l  =  0    to     ^  =  -i 

dx'  6x'  6x'  a' 

which  shows  that  the  straight  line,  from  well-known  princi- 
ples, must  cut  the  first  limiting  line  perpendicularly ;  and,  in 
like  manner,  from 

y^ax  +  h     and     -^^^^^  +  1-0, 

it  may  be  shown  that  the  right  line  must  also  cut  the  second 
limiting  line  perpendicularly. 

Supposing  the  co-ordinates  of  either  extremity  of  the  in- 
tegral, as  x'  and  y'  are  given ;  then  since  (5a?' =  0  and  Sy'=^  0, 
the  preceding  equations  will  be  reduced  to 

y  =  ax  +  h    and     ^^  +  1^0; 
consequently,  the  straight  line  must  be  drawn  from  the 


326  MAXIMA  AND  MINIMA 

point  whose  co-ordinates  are  x'  and  ij\  perpendicular  to  the 
second  limiting  line.  Hence,  when  the  limiting  lines  are 
straight  and  in  the  same  plane,  thej  must  be  parallel ;  for, 
otherwise,  the  minimum  integral  will  evidently  be  impos- 
sible. 

Eemarks. — 1.  We  have  dwelt  at  some  length  on  this 
example,  because  of  its  simplicity  and  great  use  in  showing 
how  to  find  the  variations  of  indefinite  integrals,  preparatory 
to  the  determination  of  the  forms  of  those  that,  between 
given  limits,  admit  of  maxima  or  minima  values. 

2.  If  the  limiting  lines  are  not  independent  of  each  other, 
then  the  equation,  which  expresses  their  dependence,  must 
be  noticed ;  and  thence  the  solution  may  be  obtained. 

3.  If  our  object  had  been  merely  to  find  the  nature  of  the 
integral  in  6,  it  is  easy  to  perceive  that  /  ^dj^  -\-  dy'^  might 
have  been  written  in  the  form 


/i/r7(2)'x^, 


and  the  variation  taken  by  regarding  y  as  a  function  of  a;, 
supposing  dx  to  be  constant,  or  x  to  be  the  independent 
variable,  which  would  have  led  to  y  =  ax  -{-h,  the  equation 
of  a  straight  line,  as  at  p.  324 ;  but  this  process  would  not 
have  indicated  the  manner  in  which  the  line  must  cut  its 
limiting  lines,  as  by  the  preceding  method, 

(7.)  We  will  now  proceed,  according  to  what  is  sometimes 
regarded  as  the  particular  object  of  the  method  of  variations, 
to  consider  the  maxima  and  minima  of  Indefinite  In- 
tegrals. 

1.  Let   jY  =  u  represent  the  integral  of  any  difierential ; 


OF  INDEFINITE   INTEGRALS.  827 

sucli,  that  the  form  of  /  V  in  terms  of  its  variables  is  to  be 

found,  so  as  to  satisfy  certain  maxima  or  minima  conditions  ; 
then,  since  it  is  supposed  that  the  integral  can  not  be  ex- 
pressed except  by  the  sign  T,  it  may  be  regarded  as  being 
of  an  indefinite  form. 

2.  If  u  receives  a  small  variation  or  change  of  form,  in 
consequence  of  small  changes  in  the  relations  of  its  variables  ; 
then,  from  the  generalization  of  Taylor's  Theorem,  as  at 
pp.  2 1  to  23,  by  using  6  for  d^  it  is  clear  that  it  will  become 

^  +  ^''+  12  +  1:2:3  +'^'-' 

such,  that  Su  contains  the  simple  powers  of  the  variations  of 
the  variables  in  t^,  ^hc  contains  two  dimensions  of  the  same 
variations-,  6^  it  contains  three  of  their  dimensions,  and  so  on. 
(See  Lacroix,  vol.  II.,  p.  788.) 

Hence  (see  p.  94,  &c.),  by  a  process  of  reasoning  analogous 
to  that  used  when  treating  of  the  maxima  and  minima  of 
definite  forms,  it  evidently  follows  that,  in  order  to  find4he 
maxima  and  minima,  we  must  assume 

6u  :=  6  I  Y  —  j  6Y  (between  proper  limits)  =  0  ; 

that  is,  the  coefficient  of  each  arbitrary  variation  under  the 

sign  /  must  be  put  equal  to  naught ;  noticing,  that  the  forms 

of  ?/,  which  make  6-it  negative,  correspond  to  maxima,  and 
those  which  make  it  positive,  give  minima. 

3.  By  calculating    /  SY  the  integral  of  the  variation  of  Y, 

it  will  be  found  (as  in  Ex.  5,  at  p.  322  )  to  consist  of  two 
parts ;  one  of  which,  containing  in  its  most  general  form  (or 


828  MAXIMA  AND   MINIMA 

when  tlie  variables  in  u  or  /  V  are  regarded  as  being  inde- 
pendent of  each  other)  as  many  terms  as  there  are  indepen- 
dent variables  under  the  sign  /  ,  each  of  these  terms  having 
the  variation  of  one  of  the  independent  variables  for  a  factor, 

■while  the  remaining  part  is  freed  from   /  ,  and  results  from 

taking  the  integral  of  the  variation  of  the  proposed  integral 
from  its  first  to  its  second  limit     Because  the  terms  under 

the  sign  /  are  clearly  independent  of  those  without  it,  and 
of  each  other,  it  is  manifest  that  to  reduce  /  6Y  to  naught, 
we  must  put  the  coefficient  of  each  variation  under    /   equal 

to  naught.     Hence  we  shall  have  as  many  equations  as  /  Y 

or  u  is  conceived  to  contain  independent  variables,  which,  by 
the  required  integrations,  as  in  5,  will  be  reduced  to  one  less 

in  number  than  before,  or  than  /  Y  has  been  supposed  to 

contain  independent  variables,  which   is   evident  from  the 

consideration  that  the  form  of/  V  will  be  determined. 

As  to  the  terms  of  /  (5y,  that  are  freed  from  /  ,  they  must 

be  reduced  to  naught,  and  treated  in  ways  very  analogous  to 
those  used  in  5,  at  p.  322,  &c. 

Hence,  the  manner  of  satisfying  /  6Y  =  0,  becomes  too 

evident  to  require  any  further  explanation. 

EXAMPLES. 

1.  Supposing  a  heavy  body  to  descend  from  one  point  to 
another,  not  in  the  same  vertical  line,  it  is  proposed  to  find 


OF   INDEFINITE   INTEGKALS.  829 

the  nature  of  the  line  in  whicb.  the  body  may  descend  in  the 
least  time. 

Let  X  and  ?/  represent  the  rectangular  co-ordinates  of  the 
place  of  the  body,  at  any  time  t  of  its  motion  from  the 
commencement ;  then,  if  Vd:j(?  +  dif'  =  ds  =  the  differential 
of  the  described  arc  of  the  sought  curve,  and  v  the  velocity 
acquired  by  the  body  in  its  descent,  and  dt  the  differential 
of  the  time,  we   shall,  from  well-known  principles  of  me- 

ds 
chanics,  have  di  =  -—]  or,  by  taking  the  integral  from  the 

time  of  the  body's  leaving  the  highest  point  to  its  arrival  at 

the  lowest,  we  shall  have  t  =  I —  ,  which  is  to  be 

a  minimum. 

If  ^  =:  32|-  feet,  and  y  the  vertical  descent  in  the  time  t ; 
then  (from  p.  lod  of  Young's  "Mechanics,"  or  any  of  the 
common  works  on  Mechanics),  we  shall  have  v  =  ^'Igy ;  and 
shall  thence  get 


-/ 


\^db?  +  dif- 


V'2gy 

which  must  be  a  minimum;  or,  since  2g  is  constant,  it  is 
evident  that  when  ^  is  a  minimum, 

is  also  a  minimum. 

It  is  hence  manifest  that   for  /  V  we  must  here  take 

J  y  — ~ ;  and  that  x  and  y  may  be  regarded  as  being 

independent  of  each  other. 

Since  (from  what  has  been  shown")  we  have 


330  MAXIMA   AND  MINIMA 

if  if 

we  shall  get,  by  taking  the  variations,  V2g  6t  =  0,  because 
t  is  to  be  a  minimum,  or  its  variation  equal  to  naught ; 
which  gives 


Hence,  integrating  this  equation  by  parts,  and  using  C  for 
the  arbitrary  constant,  we  shall  have 

d8\/y  ds\/y   ^  ,  J      ds\/y 

If  x\  y\  and  x'\  y'\  represent  the  co-ordinates  of  the  body 
at  its  highest  and  lowest  points,  we  shall,  from  the  elimination 
of  C  from  the  equation,  have 

ds'^y'  da"  \y"    ^         \ds'  \/y'  ds'  \/y'    ^  / 

J      d8\/y  J  \    ds  \/y       1  \/yV    *^ 

This  equation  may  be  satisfied  by  putting  the  coefficient  of 

/dr, 
equal  to  naught,  or  d  -.— ~  =^  0,  whose 

r  if 

dx  1 

integral  may  be  represented  by  -, — —  =  — -  =  the  arbitrary 

constant ;  and  because  this  equation  is  sufficient  to  determine 
the  curve  described  by  the  body,  we  may  reject 


OF   INDEFINITE   INTEGRALS.  331 

chj  1     (Is 


f{''£i^  +  2-^^'V' 


or  assume  dy  =  0 ;  and  because  the  first  and  last  points  of 
the  curve  described  by  tlie  body  in  its  descent  are  given,  we 
shall  have 

consequently,  the  above  variational  equation  is  fully  satisfied. 
To  determine  the  curve  described  from  the  equation 
dx    _    1 
ds  \/y       \f'a ' 
by  taking  its  square,  we  have 

dx^        1  dx^       y 

dsy       a  as''       a 

d'U        a ij 

which,  since  ds'^  —  dx'  -f  dif^  gives  ^  = .     Putting 

CbS  Cb 

a  —  y-^z  and  dy  =  —  dz^  this  becomes 

dz^  _  3  di  _     /z 

ds^  ~  a  ds~  ^  a^ 

which  reduces  to  dzz~^  =  -—  ;  whose  integral  ffives 

|/a  DO 

2/as  =  5     or    s^  =  4:az^ 

which  needs  no  correction,  supposing  the  arc  s  to  be  estimated 
from  the  lowest  of  the  given  points  upward,  z  being  the 
abscissa  corresponding  to  the  arc  s.  It  is  manifest,  from  what 
is  done  at  p.  150,  that  6*^  =  4as  is  the  equation  of  a  cycloid 
having  its  vertex  at  the  lowest  of  the  given  points,  and  a 
the  perpendicular  from  the  lowest  point  to  the  axis  of  a?, 
supposed  to  be  horizontal  and  to  pass  through  the  highest 
point,  for  its  axis  or  the  diameter  of  the  generating  circle ;  and 
the  point  of  the  axis  of  x  between  the  highest  point  and  the 
pei-pendicular  a,  is  manifestly  the  semibase  of  the  cycloid. 


882  EXAMPLES. 


The 

same  can 

be  shown  from    . 

dx 
~ds 

=  /! 

-^  %  =  v 

/z 
a' 

wliich 

give 

dx 

dz 

=  /i'= 

=  /-^-^. 

or 

dx^ 

adz  — 

^  ^ 
../.       2^^- 

zdz 

a 

^  versin  z 

dz 

Vaz 

-z'         Vaz- 

v. 

■  j 

az  —  z^ 

-  dV az 

—  z^  -\-  arc  rad 

whose 

integral  gives 

X  =  Vaz  —  z'-\-  arc  rad  ^    and    versin  z^ 

which  agrees  with  the  well-known  equation  of  the  cycloid, 
when  the  origin  of  the  co-ordinates  is  at  its  vertex,  and  its 
axis  is  that  of  z.     (See  p.  150.) 

Eemarks. — 1st.  If  the  body  is  to  move  in  a  vertical  plane 
from  a  higher  to  a  lower  line  or  curve,  in  the  shortest  time, 
then,  when  the  lines  are  so  placed  that  the  solution  is  possi- 
ble, it  may  be  shown,  as  in  example  5,  at  p.  322,  that  the 
cycloid  must  cut  the  limiting  lines  perpendicularly :  also,  if 
the  body  is  to  move  from  a  point  to  a  lower  line,  it  must 
move  in  a  cycloid  which  cuts  the  line  at  right  angles. 

2d.  We  may  find  the  time  of  descent  from  the  highest  to 
the  lowest  of  the  given  points,  as  follows : 

Thus,  by  taking  the  differential  of  *'  =  4a5;,  we  get 

^sds  =  ^adz     or     ds  = =-dz\/  -', 

4/3  ^  z 

consequently,  since 

V  =  Vigy  —  Vig  {a  —  z\     or    ds  =  dz  y- 
we  shall  have 


dt^  — 


TIME   OF   VIBRATION   IN   CYCLOID.  S33 

ds 


=  -/^ 


V2rj{a-2)  ^   2^  V(a5-2-)' 

by  using  —  in  the  right  member,  because  2  decreases  when 
t  increases.  It  is  easy  to  perceive  that  this  equation  is 
equivalent  to  the  form 

di  =  —  d  arc  [rad  -  versin  2]  -. — . 

T 

If  -  represents  the  time  of  descent  of  the  body  from  the 

highest  to  the  lowest  point,  then,  since  the  arc  whose 
radius  =  1  and  versin  =  2  is  tt  =  3.14159,  &c.,  =  the  semi- 
circumference,  whose  rad  =  1,  and  that  the  arc  whose 
versin  r=  0  is  also  naught,  by  taking  the  integral  of  the  pre- 
ceding differential  equation  from  the  arc  whose  versin  —  0, 

we  shall  get  ^  ==  tt  .|/  — ,  as  required. 

If  z'  is  supposed  to  be  unlimitedly  small,  the  preceding 


value  of        x'  =  Vaz  —  2''^  +  arc  (rad  -  versin z\ 

on  account  of  the  comparative  smallness  of  2'^^  and  because 

arc  rad  =  -  and  versin  z'  differs  insensibly  from  its  chord 

Vaz'^  may  evidently  be  reduced  to  x'  =  2Vaz'  very  nearly. 
Now,  if  the  arc  of  the  cycloid  corresponding  to  z'  is  represented 
by  6-',  the  equation  s^  =  4as  becomes  s^'-^  —  4-az\  whose  square 
root  gives  /=  2Vaz' ;  consequently,  when  z'  is  so  small  that 
z'^  may  be  regarded  as  an  infinitesimal  in  comparison  to  z'^ 
we  shall  have  x'  =  s'  very  nearly.     Hence,  because  .s-'-  ==  4:az' 

gives  4a  =  — r  ,  we  shall  have  4a  =  --7- ;   consequently  (from 

z  z 

a  well-known  property  of  the  circle),  4a  .equals  the  diameter 
of  a  circle  which  has  the  same  curvature  as  the  cycloid  at  its 
lowest  point,  or  vertex. 


834  TIME   IN   UNLIMITEDLY 

Because  «'  =  2  ^az'  is  sensibly  the  same  as  a  circular  arc 
whose  radius  is  2a  and  height  s',  it  clearly  follows,  from  what 
has  been  shown,  that  the  line  of  descent  down  the  (infinitesi- 
mal) circular  arc  height  z'  and  rad  =  2a,  differs  insensibly 
T 

2-'  y  2g 
From  this  equation,  we  have 


from^  =^\/^. 


which  equals  the  time  of  an  infinitesimal  vibration  of  a 
pendulum,  whose  length  is  2a.  Hence,  if  the  vertical  dis- 
tance of  any  two  points  is  denoted  by  a,  it  clearly  follows 

T  /a 

from  ^  =  ^  T  9~ '  ^^^^  ^^^  least  time  in  which  it  is  possible 

for  a  heavy  body  to  pass  from  the  higher  to  the  lower  point, 
equals  the  time  of  one  vibration  of  the  pendulum  whose 

length  is  ^ ,  or  half  the  time  of  one  vibration  of  the  pendu- 

A 

lum  lohose  length  is  2a. 

3d.  The  question  here  treated  of,  is  sometimes  called  the 
Problem  of  the  Brachystochrone,  or  the  line  of  quickest 
descent. 

2.  Two  points  in  a  vertical  plane  are  connected  by  a  line 
of  uniform  diameter  and  density,  to  find  its  nature  when  its 
center  of  gravity  is  lowest 

Let  the  line  be  referred  to  the  axes  of  x  and  ?/  in  its  own 
plane ;  the  axis  of  x  being  horizontal  and  y  vertical,  and 
directed  downward. 

Then,  ds  being  the  differential  of  the  length  of  the  line, 

I  ds  =  const  when  taken  throughout  its  entire  length,  will 
be  one  of  the  conditions  of  the  question ;  and  /  yds  =  max., 


SMALL   CIRCULAR  ARCS.  385 

when  taken  tlirougli  the  entire  length  of  the  line,  will  be  the 
other  condition,  since  this  integral,  divided  by  the  length  of 
the  line,  expresses  the  descent  of  the  center  of  gravity. 

If  we  multiply  the  first  condition  by  the  constant  a,  and 
add  the  product  to  the  second  condition,  the  two  will  clearly, 
since  the  first  is  constant,  be  reduced  to  the  single  condition, 

a  I  (is  +  I  yds  =  I  [a  -\- y)  ds  =  max. 

By  taking  the  variation,  we  have 

6  I  {a-\-y)  ds  =  j  6  {a  -\-  y)  ds 

=f{ds6y  +  (a  +  y)^6dx  +  {a  +  y)^^Sdy) 

=J  (^  +  2/)  ^  (^^^  +J\dsdy  +  (a  +  2/)  ddy-]^ 
which  integrated  by  parts  gives 

(«  H-  2/)  -J  ^^  +  («  +  2/)  J  dy  -fd{a  +  V)  £  ^^' 

-.f(^^ds  +  d{a  +  y)^)dy  +  C  =  0, 

C  being  the  constant. 

Since  the  extremities  of  the  integral  are  given  points,  the 

part  of  the  integral  without  the  sign  /vanishes,   and  the 

constant  =  0 ;  also,  since  ^x  and  6y,  under  the  sign  /  ,  are 
arbitrary  and  independent  of  each  other,  we  must  have 

^4(^^  +  2/)^}  -=0,     and     .-c4>+c^|(a+2/)^}  =  0. 

The  integral  of  the  first  of  these  gives 
do) 
(«  +  2/)  ^  =  ^  =  const., 

which  gives 


836  TIME  IN   UNLIMITEDLY 

(a  ■\-yfcb?=z  h'  {dx"  +  df),    or    [(a  -f  yf  -  J^  dx"  =  IHif, 
hdi/ 


or  dx  = 


V{a-\-yy-T'' 


and  by  putting  the  second  equation  in  the  form 


^  («  +  y)  ;^  =  d^i 


and  integrating,  we  have 


(«+2')|  =  *  +  C, 


C  being  the  arbitrary  constant. 
If  -—  =  0  when  5  =  0,  we  have 


dif 


(«+.)!= 


since  C  =  0 ;   and  thence  2ad(/  +  2(/di/  =  28dSj 

whose  integral  gives  2ay  -{- 1/  =  ^, 

which  needs  no  correction,  supposing  s  and  y  to  commence 

together  and  to  be  reckoned  upward ;  noticing,  that  the  origin 

of  the  co-ordinates  is  clearly  at  the  vertex,  since  ~  ==  0  at 

the  origin.  The  preceding  equation  is  the  common  catenary, 
the  well-known  curve,  into  which  a  uniform  chain  of  unlim- 
itedly  small,  short  links,  when  suspended  from  its  extreme 
points,  will  form  itself;  and  it  is  also  well  known  that  the 

equation  dx  =     ,     — =£:r-    , 

Sfi^a^yf^l?' 

previously  found,  is  another  form  of  the  equation  of  the 

same  curve. 

Eemarks. — Because  the  length  of  the  curve  in  this  ques- 
tion is  given  in  addition  to  the  maximum  condition,  the 
question  is  said  to  fall  under  the  class  of  what  are  called 
uojpeinmetrical  questions. 


/ 


SMALL   CIRCULAR  ARCS.  837 

3.  To  find  the  relation  between  oo  and  y,  when  the  integral 
ydf 


-^ ,  taken  between  proper  limits,  is  a  minimum. 


which,  integrated  by  parts,  becomes 

ZydyHa?  +  ydy*  iydfSx   ,   p   , 

/( j:  _  rf  ?y^<^+^J  ^^  ^y^  2jgy.  ^^  ^  ^. 

in  wbicb  •  db^  is  put  for  dd(?  +  6?y^,  and  C  is  tbe  arbitrary 
constant. 

Because  tbe  equation  must  be  satisfied  so  as  to  leave  5y 

and  ^x  under  tbe  sign  /  arbitrary,  we  must  put  their  coeffi- 
cients equal  to  naught,  and  sball  thence  get 

%  -  d^yM^^p!^  =  Q    and     dy^P^O; 

ds^  as*  ■  as* 

consequently,  the  preceding  variation  reduces  to 

Sydfd^^  +  ydy*       _  2y^'dx  ^  ^  ^^ 

ds*  ^  ds* 

If  the  extremities  of  the  integral  are  given  points,  we  have 
6y  =  0,  6x  =  0,  and  thence  C  —  0 ;  consequently,  the  con- 
ditions of  the  question  are  all  satisfied. 

To  find  the  relation  of  x  and  y,  it  will  clearly  be  sufficient 
to  take  the  integral  of 

d  ^-jj-  =  0,  which  gives  ^  J^  ^  =  C  =  const, 
15 


3o8  TIME  IN  UNLIMITEDLY 

and  to  reject  the  other  equation,  or  to  put  the  6y  under  the 
sign  /  ,  equal  to  naught 

The  equation  —fj—  =  C,  gives 

—  c'      ^^*  —  c    (^  +  ^yy 

^  ~  dy^dx~  dy^dx 

by  putting  '-£,  =  P- 

From  y  =  ^^r^  =  ^'  (^"^  2i?-^  +p\ 

we  get         c??/  =  C  (-  3j?-'*  -  2^-'  +  1)  dp, 
and  thence  c?aj  =  —  becomes 

whose  integral  gives 

-  =  o"  +  c-(|,  +  l.  +  h.i.4 

in  which  0"  is  the  arbitrary  constant,  and  h.L^  denotes  the 
hyperbolic  logarithm  of  j9. 

Supposing  ^  to  be  eliminated  from 

then,  by  putting  in  the  resulting  equation,  the  values  of 
X  and  y  at  the  given  points  at  the  extremities  of  the  integral, 
we  shall  have  two  equations  containing  C  and  C  as  un- 
knowns, whose  solutions  will  give  the  required  values  of  the 
constants,  as  required;  consequently,  the  required  relation 
between  x  and  y  will  be  found. 


SMALL   CIRCULAR   ARCS.  839 

Instead  of  supposing  the  extremities  of  the  integral  to  be 
given,  it  will  clearly  be  sufficient  to  use  other  conditions  ; 
such  as  will  enable  us  to  find  the  constants  C  and  C,  and 
thence  to  get  the  values  of  x  and  y  that  ma}^  correspond  to 
any  assumed  value  of  ^. 

Thus,  if  the  limits  of  the  variation  of  the  integral  are  not 
given  points ;  then,  if  the  variation 

is  taken  from  the  values  of  x  and  y  represented  by  x'  and  y' 
to  the  values  represented  by  x"  and  y'\  we  shall  have 

Zy"dy"Hx"'^  +  y"dy"^  .  ,,       2y"dy"Hx" 


,//i 


'y'^--^"-^^^'^" 


iZy'dy'^dx'^^y'dy''  2y'dy''dx'       \  _ 

~  V  dl'  y  d^'^~  00.  j  -i). 

If  the  co-ordinates  at  the  extremities  of  the  integral  are 
independent  of  each  other,  it  is  manifest  that  this  equatioa 
will  be  divided  into  the  equations 

which  representing  -^  by  p,  are  equivalent  to 

^__¥:_     and    ^-      2^' 
6x''  ~  3  -\-2)"'  dx'  ~~  3TF^' 

Since 

y  =  -^^'   and    .  =  C"  +  C  (A  +  ^  +  1,L^), 

we  may  to  these  join  the  equatioas 


340  TIME   IN   UNTilMITEDLY 

\9 


C'a+y"y    ,_c'(i+y°)' 
a,"^C"  +  c'(J-,;5+^  +  h.i.y'), 

and,  representing  the  equations  of  the  limiting  curves  by 
y-=cl>{x'')    and    y'='^{x'\ 

we  shall  have    ^  =  (p'  {x/')     and    ^  =  V*  (»0) 
•which  reduce  the  preceding  equations  to 

Hence,  we  have  eight  equations,  which  will  enable  us  to 
find  the  eight  unknowns,  x'\  y'\  p'\  x\  y\  p\  C,  and  C ; 
consequently,  the  points  in  which  the  curve  represented  by 
the  equations 

intersects  the  limiting  curves  y"  =  ^'  {x")  and  y  =  -^  {x'\ 
may  be  supposed  to  have  been  found ;  and  since  the  con- 
stants C  and  Qi"  may  be  supposed  to  have  been  found,  it 
clearly  follows  that  the  curve  represented  by 
01(1+^ 


and    «,  =  C"+C'(|i  +  i5  +  h.l.^.), 
may  be  supposed  to  be  drawn,  as  required,  between  its  limit- 


If  for  either  limiting  curve,  as  that  whose  co-ordinates  are 
x'  and  y\  we  take  the  point  whose  co-ordinates  are  x'  and  y\ 
then  it  is  easy  to  perceive  that  our  eight  equations  will  be 


SMALL   CIRCULAR  ARCS.  341 

reduced  to  six,  whicli  will  enable  us  to  find  ttie  six  un- 
knowns, x'\  y'\  J)" ^  p\  C,  and  C^^,  &c.,  as  before. 

Remarks. — 1.  It  is  easy  to  perceive,  from  tlie  solution  of 

Ex.  19,  at  p.  113,  that  /  -j-/         ^  represents  the  resistance  of 

a  solid  of  revolution  around  the  axis  of  a?,  moving  in  a  fluid 
of  uniform  density,  in  the  direction  of  the  axis  of  x  with  its 
smaller  end  foremost,  whose  nature  we  have  determined,  so 
as  to  make  the  resistance  a  minimum. 

2.  The  example  is  substantially  the  same  as  that  solved  by 
ISTewton,  at  p.  120,  vol.  II.,  of  his  "Principia."^  If,  in  the 
preceding  equations,  we  put  ^  =  1,  and  y'  for  the  corre- 
sponding value  of  y  and  x  =  0,  then 

2/' =40',     or    C'  =  |,    0  =  C"  +  ~C',    or   0"=-^. 

From  the  substitution  of  the  values  of  the  constants,  the 
equations  become 

2/  -  4        ^^,      ,     ^nd    X-  ^\^^^,  +  ^,  +  hA.p       ^  j  , 

which  clearly  reduce  to  y^  and  0  at  the  origin  of  the  co-ordi- 
nates, since  h.  1. 1  =  0.  If  we  put  ^  =  0.9,  we  readily  get 
y  =  1.1232/'  ^^^  *'  —  0.130 y'  very  nearly,  and  p  —  0.8 
gives  y  =  1.313?/'  and  x  —  0.354?/'  nearly,  and  so  on. 

Hence,  when  the  extremities  of  the  integral  are  fixed 
points,  as  at  p.  337,  we  easily  perceive  how  the  equation 
which  connects  y  and  x  may  be  represented  by  linear  de- 
scription. 

Thus,  by  putting  2/^=1,  and  assuming  ox  and  oy  for  the 
positive  directions  of  the  rectangular  co-ordinates,  having  o 
for  their  origin,  we  set  1  from  o  on  the  axis  of  y  for  a  point 


842 


TIME  IN  UNLIMITEDLT 


in  the  curve,  and  then  set  0.130  from  o  on  the  axis  of  a?, 
through  which  (point)  we  erect  the  perpendicular  1.123  to 
the  axis  of  x  for  the  corresponding  value  of  ?/,  and  then 
having  set  0.354  for  x'  from  o^  as  before,  we  draw  the  per- 
pendicular through  the  point  to  the  axis  of  x  equal  to  1.313 
for  the  corresponding  value  of  y,  and  so  on  to  any  required 
extent;  then,  a  curve  drawn  with  a  steady  hand  through 
the  points  thus  found  will  be  such,  that  by  revolving  around 
the  axis  of  x  it  will  generate  a  solid,  which,  moving  in  a  fluid 
from  X  toward  o,  it  will  meet  with  less  resistance  than  any 
other  solid,  whose  end  diameters  and  height  are  the  same. 
It  is  manifest,  that  the  preceding  construction  is  substantially 
the  same  as  that  of  Newton. 

4.  To  find  the  curve  surface,  whose  area  between  given 
limits  is  a  minimum. 

Agreeably  to  what  is  shown  at  p.  302,  the  double  integral 

when  taken  between  the  proposed  limits,  may  be  taken  to 
represent  the  required  surface. 

By  taking  the  integral  of  the  variation  of  the  surface,  we  have 


SMALL   CIRCULAR  ARCS.  343 

by  using  'p  and  ^  for  —  and  ~,  and  because  z  is  regarded  as 

being  a  function  of  x  and  y,  considered  as  being  independent 
variables.     Since 

dp  z=  o  —  =  —-     and     on  ^=  -=- 

^         dx        ax  ^       dy 

on  account  of  tbe  constancy  of  dx  and  dy ;  tben,  if 
-g and    Q 


^'  =^ff^^y  -£  ^^  +/7q^^  ^  ^y- 


we  sliall  bave 

Hence,  integrating  by  parts,  we  sball  have 

6h  ^fvdySz  +fQdxd3  -ff^^  ^zdxdy  -ff^^  ^^dxdy ; 

and  it  is  clear  tbat  tbe  part  of  this  integral  wbicb  is  freed 
from  one  of  the  signs  of  integration,  since  it  relates  to  tbe 
fixed  limits,  must  be  reduced  to  naught,  since  Sz  at  the 
limits  =  0.     Hence,  we  shall  have 

^'^-fJ%,^^^''^y-ffdy^^^''^^^  • 

which  must  equal  naught,  since  s  is  to  be  a  minimum  ;  con- 
sequently, since  dz^  under  the  double  sign  of  integration,  is 

indeterminate  or  arbitrary,  its  factor  -7-^  +  -^ ,  under  the 

double  sign  of  integration,  must  be  reduced  to  naught,  which 

gives  -7 [-  -—  z=  0.     By  restoring  the  values  of  P  and  Q, 

dx        dy 

and  taking  the  indicated  differential  coefi&cients,  the  preceding 

equation  will  be  reduced  to  its  equivalent, 


344  TIME  IN  UNLIMITEDLY 

consequently,  if'  we  put 

f^r,    ^  =  ^  =  *,    and    ^  =  t, 
ax  ax       dy  ay 

we  shall  have 

(1  +  q')r-2jpqs  +  {l+p')t  =  0, 

for  the  equation  of  the  partial  differential  coefficients  of  the 
sought  surface,  whose  integral  will,  of  course,  be  the  surface. 
Remarks. — 1.  This  example  has  been  taken  from  p.  753, 
vol.  II.,  of  Lacroix's  work,  where  it  is  remarked,  in  a  foot 
note,  that  the  equation 

^_.   ^  =  0       ves    ^=:  _^- 
dx        dy  °  dx  dy  ^ 

which  is  the  condition  of  the  immediate  integrability  of 
Vdy  —  Qfitx ;  consequently,  it  is  concluded  that  on  the  mini- 
mum surface,  —TTT^- — t"- — 17  is  an  exact  differential,  as  well 
.'.|/(l+y  +  ^-0     - 

as  dz  =pdx  +  qdy.  Thus,  all  plane  surfaces  will  be  found 
to  satisfy  these  conditions ;  since 

and  of  course  the  preceding  conditions  are  reduced  to 
naught;  consequently,  since  the  differentials  of  constants 
equal  nought,  it  is  manifest  that  the  preceding  differentials 
may  be  regarded  as  having  constants  for  their  exact  in- 
tegrals. 

If  these  tests  are  applied  to  the  surface  whose  equation  is 
az  =  xij,  they  will  be  found  to  give 

_dz  _y         ,         _  dz  _x 
■^  ~  dx~  a  ^  ~  dy''  a"* 

which  reduce  them  to 


SMALL  CIRCULAR  ARCS.  845 

ydy  —  xdx  j     j    _  2/^^-^  +  •'^^//  _  ^^  {^V) . 

4,/  (a?  +  x^  -\-  y^)  "  a  a      ' 

consequently,  since  the  first  of  tliese  is  not  an  exact  differen 
tial,  it  follows  that  the  proposed  surface  does  not  belong  to 
the  class  of  minimum  surfaces.     Nevertheless,  if  x  and  y  are 

very  small  in  comparison  to  a,  it  is  clear  that  -yj-rf- — i        2\ 

ijcLij xdx 

does  not  sensibly  differ  from  —- ,  which  is  an  exact 

differential;  consequently,  if  a  is  very  great,  the  surface 
az  —  xy  for  finite  values  of  x  and  y,  will  not  greatly  differ 
from  a  minimum  surface. 

2.  Lacroix,  at  pages  806  and  875  of  the  volume  cited, 
shows  how  to  find  the  solid  which,  with  a  given  capacity, 
contains  the  least  surface. 

Thus,  since  J  J  zdxdy     and    JJ  ^  (1  +i?^  +  2'')  ^^^y 

express  the  capacity  and  surface,  and  that  the  first  is  given, 
it  is  manifest  if  0  stands  for  a  constant,  that  when  the  sur- 
face is  a  minimum, 

ffCzdxdy  ^ff  V{\  +i?^  +  ct)  dxdy  ■= 

fJ\C3  +  V{^+p'-\-r)\dxdy 

will  also  be  a  minimum.  Hence,  using  P  and  Q  to  stand 
for  the  same  things  as  before,  then  taking  x  and  y  for  the 
independent  variables,  we  in  like  manner  get  the  equation 

ax       ay 
or  its  equivalent, 

0  (1  +y  +  ff  -  [(1  +  ct)  r  -  2/?^^  +  (1  +/)  t\  =  0, 

for  the  equation  of  partial  differential  coefficients  of  the  re- 

15* 


34:6  TIME  IN  UNLIMITEDLY 

quired  body,  whose  integral  will,  of  course,  represent  tlie 
body. 

Lacroix  remarks,  that  the  sphere  and  cylinder  whose 
equations  are  represented  by 

2'  -hf-\-x'  =  a'^,     and     z- +  y- =  a'\ 
will  satisfy  the  preceding  equations. 

Thus,  in  the  sphere    ^  = ,     and     q  z=i  —  ^^ 

which  give  |/(1  +  p-  -\-  q^)  =  -^ 

z 

and  thence    P  and  Q  equal and    —  - :   which  reduce 

^  ^  a  a 

2 
the  first  of  the  preceding  equations  to  C  +  -  =  0,  and  in  the 

cylinder  the  same  equation  is  reduced  to  C  H — >  ==  0. 

5.  To  draw  the  shortest  line  possible  from  one  point  to 
another,  on  any  proposed  surface. 

Let  a?,  y,  2,  represent  the  rectangular  co-ordinates  of  any 
point  of  the  sought  line ;  then,  because  the  point  is  on  a 
surface,  z  may  be  considered  as  being  —  a  function  of  x  and 
y  regarded  as  being  independent  variables,  and  we  shall  have 

T        ch   .,        dz  ^         ' 

Tx       '^  dy  ^y  "^  ^      ^  ^  '^' 

From  what  is  done  at  p.  240,  we  evidently  have 

ds  =  x/{dx'  +  dy''  +  dz') 

for  the  differential  of  the  line,  and 

s  =  f^{dx'  +  dy'  +  dz^) 

will  represent  the  line,  and  its  variation  becomes 


SMALL   CIRCULAR  ARCS.  347 

ds  =  df^/idx^  +  dy^  +  dz'')  =z  Cd  .^{d'j?  4-  dy"  +  d^) 

dx   ^  dy    p  d^   , 

C  being  the  constant     Taking  the  integral  from  x\  y\  z\  to 
x'\  y'\  z"^  the  constant  will  be  removed,  and  we  shall  have 

Supposing  the  extremities .  of  the  integral  to  be  fixed 
points,  the  part  of  the  integral  without  the  sign  /  will 
vanish;  and  since  (5s  =  0,  we  must  have 

since  ^z  =  p6x  +  qdy. 

Because  dx  and  dy^  under  the  sign  /  ,  are  arbitrary,  their 

factors  must  be  put  equal  to  naught,  which  give 

^  dx  ^(Iz        -  ,      ,  dii  ^  dz        - 

a  -, — \r  pa  -z-  ^  \)     and     a  -^  4-  <7«  -r-  =^  ^  I 
ds       ^     ds  ds       ^    ds 


848  TIME  IN  UNLIMITEDLY 

wLich  are  the  equations  of  the  minimum  line,  and  are  tlie 
same  as  those  given  by  Lacroix,  at  p.  270  of  the  volume 
befoi-e  cited. 

Thus,  to  draw  the  shortest  line  possible  from  one  point  to 
another  on  the  surface  of  a  sphere  whose  equation  is 
2^=  a^^  f+  z\ 

Here  dz  = ■  dx  —  -  dy^ 

z  z 

which  gives       ^  = and    q  =  —  -^^ 

z  z 

which  reduce  the  preceding  equations  of  the  minimum  to 

whose  integrals  may  be  expressed  by 

zdx  +  xdz  =  Ads^     and    zdy  —  ydz  =  Bds. 
Multiplyftig  the  first  of  these  by  B  and  the  second  by  A,  we 

X  1/ 

readily  get  Bd  -  =  Ad  - ; 

z  z 

whose  integral  gives 

^  +  C  -  B  -  =  0,     or    Ay  -Bx  +  Cz  =  0', 

z  z  ^ 

which  is  the  equation  of  a  plane  passing  through  the  center 
of  the  sphere,  and  of  course  the  shorter  of  the  arcs  of  a 
great  circle  which  passes  from  one  of  the  given  points  to  the 
other,  is  the  required  minimum  distance. 

Remark. — Besides  the  minimum  thus  determined,  which 
may  be  called  the  absolute  minimum  on  the  spheric  surface, 
there  is  what  may  be  called  the  relative  maximum.  For  the 
lesser  arc  of  the  great  circle,  between  the  points  being  a 
minimum,  the  remaining  arc  of  the  same  great  circle  will  be 


SMALL   CIRCULAR  ARCS.  849 

the  greatest  distance  on  the  surface  between  the  points ;  sup- 
posing the  distance  to  be  measured  in  planes  passing  through 
the  points. 

(8.)  We  may  now  proceed  to  show  how  to  distinguish  be- 
tween the  maxima  and  minima  in  examples,  but  shall  refer 
for  this  to  Art.  876,  p.  807,  of  the  "  Calcul  Integral"  of 
Lacroix;  noticing,  that  the  maxima  and  minima  can  often 
be  distinguished  from  each  other  by  the  nature  of  the  case, 
as  in  the  examples  which  have  been  given. 

As  we  do  not  profess,  in  what  has  been  done,  to  have 
given  any  thing  more  than  the  first  principles  of  the  Calculus 
of  Variations,  we  must,  for  more  ample  details,  refer  to 
larger  works :  such  as  "Woodhouse's  "  Treatise  on  the  Calcu- 
lus of  Variations,"  and  the  "  Calcul  Integral"  of  Lacroix, 
at  p.  721.     (See  p.  614,  Appendix.) 


SECTION  III. 

INTEGRATION  OF  RATIONAL  FUNCTIONS  OF  SINGLE  VARIA- 
BLES, MULTIPLIED  BY  THE  DIFFERENTIAL  OF  THE  VARI- 
ABLE. 

(1.)  It  is  clear  that  sucli  differentials  must  be  of  one  of 
tlie  two  forms 

(Aaj«  +  Baj*  -f  Cx'  +  &c.)  dx, 

A.T"  +  Bx^  +  Cx'  +  &c.      , 
A'x""  +  B  V  +  G'x'  +  kc.       ' 

in  which  tne  indices  of  x  are  supposed  to  be  positive  in- 
tegers. Supposing  the  terms  of  these  expressions  to  be 
arranged  according  to  the  descending  or  ascending  powers 
of  a?,  we  may  suppose  the  index  of  the  highest  power  of  x  in 
the  numerator  of  the  fractional  form  to  be  less  than  the 
index  of  the  highest  power  of  x  in  the  denominator ;  for  if 
the  index  of  x  in  the  numerator  is  equal  to  or  greater  than 
in  the  denominator,  it  may  be  made  less  by  arranging  the 
terms  of  the  numerator  and  denominator  according  to  the 
descending  powers  of  x,  and  then  dividing  the  numerator  by 
the  denominator,  when  the  fractional  form  will  be  reduced 
partly  or  wholly  to  the  first  of  the  preceding  forms,  accord- 
ingly as  the  numerator  is  not  or  is  exactly  divisible  by  the 
denominator. 

(2.)  By  proceeding  as  in  (9.)  at  p.  26G,  we  may  clearly 
suppose  the  integrals  of  all  such  differentials  as  the  above  to 


EATIONAL   FUNCTIONS.  851 

be  found  to  any  degree  of  exactness  that  may  be  required  ; 
whicti  is  clear  from  the  circumstance  mentioned  by  Newton, 
tliat  they  have  the  sums  of  the  inscribed  and  circumscribed 
rectangles  for  their  less  and  greater  limits. 

(3.)  If  we  have  differentials  of  the  preceding  forms,  in 
which  the  indices  of  oc  are  some  of  them  positive  fractions, 
by  reducing  the  indices  to  their  least  common  denominator, 
and  representing  unity  divided  by  the  least  common  denomi- 

nator  by  - ,  and  putting  y  =  jtp  ,  ox  x  =  y^^  we  shall  have 

1 
dx—pyP-^dy\  consequently,  putting  y  for  a?^  and  j9?/^~^<iy 
for  c?j?,  the  expressions  will  be  changed  into  forms  like  to 
those  at  first  supposed ;  and  of  course  the  integrals  may  be 
found  to  any  degree  of  exactness,  as  before. 

It  is  evident,  if  the  first  differential  has  any  negative  ex- 
ponents, that  their  integrals  may  be  found  in  algebraic  forms, 
excepting  when  any  of  them  happen  to  be  —  1,  when  the 
corresponding  integral  will  be  the  hyperbolic  logarithm  of  x 
multiplied  by  the  corresponding  coefficient  of  a?  ~  ^ ;  and  it  is 
clear,  that  if  in  the  fractional  differential  any  of  the  indices 
of  X  are  negative,  they  may  be  removed  by  multiplying  the 
numerator  and  denominator  by  x  with  the  same  index  taken 
with  the  positive  sign,  when  it  will  follow,  as  before,  that 
the  integral  can  be  found  to  the  same  degree  of  exactness, 
in  the  same  manner  as  before. 

(4.)  It  is  manifest  that  the  factor  of  dx^  in  the  fractional 
differential,  can  be  conceived  to  have  been  obtained  from  the 
addition  of  simpler  fractions  together,  after  having  reduced 
them  to  a  common  denominator ;  consequently,  the  denomi- 
nator will  represent  the  common  denominator  of  the  frac- 
tions whose  sum  equals  the  proposed  fraction.^    Hence,  to 


352  RATIONAL  FUNCTIONS 

find  the  component  fractions,  the  first  thing  to  be  done  is  to 
resolve  the  denominator  of  the  given  fraction  into  factors, 
which  can  be  done  as  follows. 

Thus,  by  putting  the  proposed  denominator  equal  to 
naught,  we  shall  have  an  algebraic  equation,  whose  roots, 
both  real  and  imaginary  (when  the  equal  roots  are  included), 
will  equal  the  number  of  units  in  the  greatest  exponent  of  x ; 
noticing,  that  the  imaginary  roots  always  enter  the  equation 
in  pairs  of  such  forms,  that  the  product  of  every  two  factors 
which  give  these  roots  will  be  real,  or  freed  from  their 
imaginary  parts. 

Hence,  we  may  suppose  the  denominator  of  the  proposed 
fraction  to  consist  of  real^  simple^  and  quadratic  factors, 
(See  p.  440  of  my  Algebra,  or  most  bf  the  common  works  on 
that  science.) 

(5.)  Having  resolved  the  denominator  into  its  factors,  and 
taken  any  one  of  its  unequal  simple  real  factors  for  the  de- 
nominator of  any  one  of  the  component  fractions,  then  we 
may  assume  a  constant,  to  be  found  from  the  principles  of 
identity  of  equations,  for  the  numerator;  since  x  must  be 
of  less  dimensions  in  x  in  the  numerator  than  in  the  de- 
nominator of  the  fraction. 

To  find  the  numerators  of  single  quadratic  factors  taken 
for  the  denominators,  they  must  generally  consist  of  a  con- 
stant term,  and  another  constant  for  a  factor  of  the  simple 
power  of  a?,  observing  that  these  constants  are  to  be  found, 
as  before,  on  the  principles  of  identity  of  equations. 

When  the  proposed  denominator  contains  a  real,  simple,  or 
quadratic  factor  in  times ;  then,  if  the  numerators  of  the 
proposed  fractions  contain  suitable  dimensions  in  a?,  the 
fraction  to  be  assumed  must  contain  the  mih  power  of  the 


OF   SINGLE  VARIABLES.  853 

simple,  real,  or  quadratic  factor,  and  may  contain  all  the 
lower  powers  of  the  same  denominator  for  the  denominators 
of  other  fractions,  down  to  the  simple  or  first  power  inclu- 
sive, provided  x  has  suitable  dimensions  in  the  numerators 
of  the  proposed  fractions ;  noticing,  if  x  does  not  enter  the 
numerator  of  the  proposed  fraction,  that  the  fraction  can  not 
admit  of  any  further  reduction. 

(6.)  To  illustrate  what  has  been  done,  take  the  following 
simple 

EXAMPLES. 

2aj 5 

1.  To  integrate  the  fraction  —^ — ~ dx. 

XT  —  k)X  -\-    O 

Putting  the  denominator  equal  to  naught,  we  have  the 
quadratic  equation  a?^  —  5a?  +  6  ==  0 ;  whose  solution  gives 
a?  =  2  or  a?  =  3,  and  of  course  a?  —  2  and  a?  —  3  are  the 
factors  of  the  proposed  denominator. 

Hence,  agreeably  to  what  is  showm,  we  assume  the  pro- 
posed fraction  equal  to 

-^-  +  ^~ 

a?  —  2       a?  —  3' 
and  thence  get  the  identical  equation 

2a^-5      _     A  JB_  _  (A  4-  B)  a?  -  3A  -  2B 

a?^  —  5a?  +  6  ~  a?  —  2       a?  —  3  "  a?"^  —  6a?  -^  6  ' 

which  gives  2a?  —  5  =  (A  -f  B)  a?  —  3 A  —  2B, 
which  must  be  an  identical  equation;  consequently,  from 
equating  the  coefficients  of  like  powers  of  x  in  the  members 
of  the  equation,  we  get  the  equations  A  +  B  —  2  and 
3A  +  2B  3=  5,  which  give  A  —  1  and  B  =  1,  and  thence 
the  proposed  differential  is  reduced  to 

2a?  —  5        ,            dx  dx 

dx  = + 


a?'  —  5a!  +  6  x  —  S       a?  —  2 


854  RATIONAL  FUNCTIONS 

whose  integral  is  expressed  hy 

/^a?  —  5       ,    _  r    dx  r    dx     _ 

^^  _  5;^.  +  6  '^''  -J.  x-Z^  J  x-^' 
log  (.-r  -  3)  -h  log  {x  -  2)  +  log  C  =  log  C  (aj  -  3)  {x-  2), 
log  C  being  the  arbitrary  constant  (see  p.  255) ;  by  putting 

C  =  ^,  we  have  y  ^r^siTe  ^  =  ^"^ 6 ' 

which  commences  with  x. 

2.  To  integrate  j^ — -^    and 


{l+xf    "  ^    (1 +«')'• 
Because  the  denominator  of  the  first  of  these  differentials 
consists  of  two  equal  factors,  1  -{-  x  and  1  +  a?,  we  assume 
X  A  B  A  +  B  +  B.27 


+ 


(1  +  xf  ~  {i-{-xy  '  1  +  x        (1  +  xf   ' 

which  gives    A  =  —  B     and    B  =  1 ;     consequently,  we 
shall  have 

/xdx  r     dx  r  dx  1         1      /-,       s     ^ 

Because  x  does  not  enter  into  the  numerator  of  the  differ- 

dx 
ential  ^^j r^ ,  we  do  not  have  any  reduction  like  the  pre- 
ceding, and  thence  immediately  get 
dx  1 


/, 


(1  +  xf  2  (1  +  x) 

for  the  required  integral 

Kemark. — The  reason  for  assuming 

X      .  A  B 


+  0 


(l+xf  {l  +  xf    '    1  +  a? ' 
becomes  evident  from  dividing  x  by  x  +  1,  which  gives 

^   =1      1 


X  +  1  1  +0?' 


OF   SINGLE  VARIABLES.  355 


and  thence  we  shall  have 

X  1 


+ 


(1  +xf  (1  +  xf    '    1  +  a^' 

agreeably  to  the  above  assumption ;  and  it  is  clear  that  a 
similar  reasoning  will  be  applicable  in  all  analogous  cases. 

3.  Integrate    2   /  :r— — §  =  log  C  (1  +  x%    and   reduce 

«/       X     ~x~    X" 

a  +  hx  -^  cx'^   ,    ^.  .      ^  „  P     •  ■  - 

ax  the  expression  to  a  propei'  lorm  tor  mte- 


/ 


(*  -  eY 
gration,  in  order  to  get  the  true  answer. 

Thus,    by  putting  x  —  e=-  z   or   a?  =  (^  -f  ;3,  dx^=.  ds,    the 
form  is 


*a  +  he  +  ce-   ,         Clre  ,  r  c 

— _ — ,i, +y  _  </, +y  ^  „ 

a  +  Z><?  +  G<i~       ^ce      ,      ^ 


as  required,  C  being  the  constant. 

A      ry^      '    .  1x—ll 

4.   10  mteorate  —. r— i ^  ax. 

^         x-^  —  'Ix'  —  .T  +  2 

Here,  because  the  factors  of  the  denominator  are  evidently 
a?  —  1,  a?  +  1,  and  x  —  %  we  assume 

7-^-11  A  _R_       __C 


x^  —  2a?-  —  a?4-2  x  —  \  x  -\-  \  a?  —  2' 
from  which,  as  heretofore,  by  the  method  of  undetermined 
coefficients,  we  may  easily  find  the  values  of  A,  B,  and  C  : 
we  will  here,  however,  use  a  modification  of  the  method, 
which  will  often  be  preferable.  Thus,  to  find  A,  we  may 
suppose  a?  —  1  to  differ  insensibly  from  naught,  which  reduces 
the  assumed  equation  to 

Tx  ~  11         __  _A_ 
^^  _  2x^  —  X  -{-  2  ~  X  —  1' 
on  account  of  the  comparative  small  ness  of  the  other  terms. 


356  RATIONAL  FUNCTIONS. 

Dividing  the  denominators  of  this  by  a?  —  1,  we  have 
7x-U     _ 
a^-x-2  ~     ' 
in  which  we  must  for  x  put  1,  which  gives  A  =  2.     By  put- 
ting a?  -f  1  =  an  infinitesimal,  we,  in  like  manner,  have 
7a; -11         _      B 
x'  —  ^x"  —  X  +2  ~  x~+  1' 
whose  denominators,  divided  bj  a?  +  1,  give 

-i^il--B 
x^-Sx-^2~     ' 

in  which  we  must  for  x  put  —  1,  which  gives  B  =  —  3 

and  in  much  the  same  way  we  get  C  =  1.     Hence 

7a?  — 11  _    2dx  Sdx  dx 


a^  —  2x'  —  x  +  2  X—  1       aj+l'a;  —  2' 

whose  integral  gives 


/ 


7.y-ll 
a;.3_  2x^-  x\-2 


=  2  log(aj  —  1)  —  3  \og{x  +  1)  +  log  (a;  —  2)  +  log  C 
(;e-Vf{x-2) 

Remark. — The  preceding  method  is  clearly  the  same  as 
to  multiply  by  a?  —  1,  divide  the  numerator  and  denominator 
in  the  first  member  by  a?  —  1,  and  put  «.■  —  l==Oora?  =  l 
in  the  result,  which  will  give  the  same  value  of  A  as  before  ; 
and  in  like  manner,  by  multiplying  by  a?  +  1  and  x  —  2 
successively,  We  get  B  and  C,  the  same  as  above. 

5.  To  integrate  ^|-^*-^^d«    and     |^- J  ^*. 
Assuming.  * 

S-x  A  B  C  D 


(x-2flx-o)  ~  {x-2f  ^  {x-2f  ^  X  -  2  ^  X  -  6' 
and  supposing  a?  —  2  to  be  an  infinitesimal,  we  shall,   on 


OF   SINGLE   VARIABLES.  '857 

account  of  the  comparative  magnitudes  of  the  terms,  have 
3  —  X  __  A ^ 

or,  dividing  tlie  denominators  of  these  hy  {x  —  2)'^,  we  liave 

3    rp 

=  A,  which,  since  x  —  2  =  0,hj  putting 2  for  a?, gives 

X  —  o 

A  =  —  g .     Hence,  b j  subtracting 
o 

A  11 


{x-2f  3   {x-2f 

from  the  members  of  the  assumed  fractions,  we  have 
S-x  1  2  1 


ix-2y(x-5)  '   3{x-2f  S{x~-2f{x-6) 


{x-2y    '    x-2    '    X  -6' 

consequently,   proceeding  with  this  in  the ,  same  way   as 
before,  we  shall  have 

B  =  —  - or     (since  x  =  2),  B  =  rr. 

3  X  —  6  ^  ^  9 

2 
Subtractinof  p—, -r-^  from  the  members  of  the  preceding 

°  9  (x  —  2f  ^ 

equation,  we  get 

2  12        1  2  1 


3{x-2f{x-6)       d  {x-2f  d  {x-2)(x-5) 

_      C  D 

~  x-2'^  x-6' 
Hence,  as  before,  we  ^have 

^="1^    °'    (smce»  =  2)  0=|.; 

2 
consequently,    subtracting  — -; —   from    the  preceding 

Z  t   [X  —  z ) 

equation,  we,  as  before,  get 


>58   •  RATIONAL  FUNCTIONS 

2  1  2  2        1 


9  {x  —  2){x  —  5)       27  (^  —  2)  27  »  —  5 ' 

whicli   must   equal  the  remaining  fraction,  and,  of  course, 

Hence,  from  the  substitution  of  the  preceding  values,  the 
integral  becomes 

J  {x-^f  {x-  5)  "^ ~  6F^'  ~  9  S=:2  "^  ^""^  ^  l;zj-6/    ' 
In  like  manner,  we  have 

(2-^y_       1  B  c 

(3  -  a;)«       (3  -  ir)^  "^  (8  -  a?/  "^  3  -  a;' 

which  gives  B  =  —  2   and  C  =  1 ;  consequently,  we  shall 
have  the  integral 


/ 


6.  To  find  the  intepral  of  7 -. . 

°  {x  —  ay 

Here  we  assume 

x^  A  B  C_         B_ 


(ic  —  of       {x  —  ay       (a?  —  af       {x  —  af       x  —  a 
or        x''  =  A-\-B(x  —  a)  +  0(x  —  af  +  'D{x  —  af, 
which  must  clearly  be  an  identical  equation ;  consequently, 
putting  a  for  x,  we  get  A  =  a\  and,  taking  the  differential 
of  the  members  of  the  equation  after  dividing  by  dx,  we 

have  3a;=  =  B  +  2C  (a;  —  a)  +  3D  (a;  -  af, 

which,  by  putting  a  for  x,  reduces  to  B  =  Sa\  By  taking 
the  differentials  of  the  members  of  -the  preceding  equation, 
we  have,  after  dividing  by  dx,  6aj  =  2C  +  6D  (a?  —  a),  which. 


FRACTIONAL   FORMS  859 

by  putting  a  for  x^  gives  C  =r  3a ;  and  taking  the  differen- 
tials again  and  proceeding  as  before,  we  bave  D  =:  1. 
Hence,  we  shall  have 

Qi?clx 


/, 


{x  —  df 
a'  3a'  3a  T      .  .       ^ 

Eemark.— -The  method  here  used  for  finding  the  value 
of  A,  B,  &c.,  appears  to  be  of  remarkable  simplicity,  and 
can  clearly, be  applied  in  all  analogous  cases. 

Othervnse,  and  more  simply. — Put  x—a  =  z  or  a7=s  +  a  ; 
then,  since  a?  =  s  -|-  a,  we  have  dx  =  dz,  and  thence 
xhlx 


/>  '     '   .,  is  reduced  to 
(x  —  a^) 


r{z  +  dfdz  _    r/(7z       Sa^^^       Za-dz        (^dz\ 
J  z^  ~~  J  \z   ^'z""^^ '^     z^      "^     ^J 


3a  _  3a'        a^ 


,  OU/  Oct  u,  ^ 

log  2  _  _-  _  _  _-^  +  C. 


(7.)  To    complete   the  integration   of  rational  fractional 

differentials,  it  clearly  follows  from  -what  has  been  done, 

that  it  is  necessary  to  reduce  the  integrals  of  differentials 

dz 
of  tbe   form    j-^. j^—,  in  which  m  is  a  positive  integral 

{z^  +  h-f  ^  ^ 

greater  than  unity,  to  that  of  like  form  in  which  m  —  1 
Thus,  from 

dz  z^dz  Irdz 


{z^  +  hy-'      {z'  +  bY      {s'  +  hy 
z  _  dz  2  (?7i  —  1)  z^dz 

by  eliminating    ^^,^_^ly    we  get 


860  FRACTIONAL   FOllMS. 

(27/2-  -  3)  dz  z_ 2{m~  \)lMz 

or  dividing  by  2  (?/i  —  1)  h^^  and  taking  the  integrals  of  the 
quotients,  we  have 

z 2;m  —  3       r         dz 

2  {ni  -  1)  <^-^  (3-  +  //)"'-^  "^  2  (//2  -  i)W  (?T-^'r~' ' 

which  reduces  the  proposed  integral  to  that  of 

dz 


f ^ 


and  by  changing  7n  into  m  —  1,  we  may  in  like  mannor  re- 
duce the  integral 

and  so  on  to  the  integral  of 

/-^ rr. ,      which  equals      -j  tan  "^  -7  ; 
z'  -{-  0-      '  *  0  0 

consequently,  all  the  preceding  integrals  pan  be  found,  as 
required. 

Thus,  if  J  =  1  and  m  =  2,  we  shall  have 

C  being  the  arbitrary  constant     Also,  if  ^  =  1  and  ?ji  —  8 
we  shall  have 

r_j}l__    _^ 3  r     dz 

J  {i-  +  I)-'  ~\{z--\-  Vf  "^  iJ  {z'  +  i)--' ' 

which,  from 


/, 


FRACTIONAL   FORMS.  861 

dz  Z  " 


(^•2  +  1)-       2(s^-  +  1)'       2 
is  reducible  to 


+  ^tan-^s, 


/ 


dz  z  ,  3s  ,    S  0.         1      ,    n 

+  TTT-^— .-^  +  77  taa-^5J  4-  0. 


{z'  +  Vf       4(^^  +  1)^    '    8(^^  +  1) 

* 

Otherwise. — Supposing  the  integral 

dz 

f  dz    _    r     J      _ite„_,f 

-^n^  +  p) 

to  commence  with  z,  then,  by  taking  the  differentials  of  the 
members  of  the  equation,  regarding  h  alone  to  be  variable, 
we  evidently  get 

-  ^^'^^fj¥T¥f  =  -  ?  *^""i  +  J  '^  ^^'^^  I ; 

or  smce  d  tan-^  n= 72--^-ll  +  pi, 

by  substitution  and  dividing  the  members  of  the  resulting 

equation  by  —  %db^  we  shall  get,  after  adding  a  constant, 
for  correction, 

dz  1 


/, 


tan-'y  + 


{z^-\-V'f~W  h    '    W{z'-\:¥)    ' 

It  is  evident  that,  by  taking  the  differentials  of  the  members 
of  this  equation,  regarding  h  alone  as  variable,  we  may,  in 

/dz 
-r-2 — —j^i  5  s^nd  so  on  to  any 

extent  that  may  be  required. 

(8.)  From  what  is  said  at  p.  851,  it  is  clear  that  if  the  dif- 
ferential of  a  variable  contains  terms  which  are  affected  with 
positive  fractional  exponents  when  the  differential  is  of  an 
integral  form,  or  positive  and  negative  exponents  when  the 


16 


3G2  FRACTIONAL   FORMS. 

differential  is  of  a  fractional  form,  that  the  differentials  m^y 
be  changed  into  others  in  which  the  exponents  shall  be  posi- 
tive integers,  or,  as  is  usually  said,  the  expressions  may  be 
rationalized. 

Thus,  the  differential  {ax^  +  ^a?  )  dx^  Avhich  is  of  an  in- 
tegral form,  by  reducing  the  indices  of  x  to  a  common  de- 
nominator, is  equivalent  to  {ax^  +  hx^)  dx ;  which,  by  put- 
ting X  =  s®,  and  dx  =  ^^^dz,  is  reduced  to  the  integral  form 
63^  {az^  +  hz^)  dz,  which  is  rationalized,  or  the  exponents  of 
z  are  integers.  By  taking  the  integral  of  the  transformed 
differential,  we  shall  have 

f{6az'^  +  Qbz")  dz  =  ^  az'  +^hz'  +  G; 

or,  putting  for  z  its  value  a?^,  we  have 

J  acc^  +  3  hx^  +  C, 

for  the  integral;  C  being  the  arbitrary  constant 

Also,  the  integral  / -r-- — 7  ™^7  clearly  be  rationalized 
*/  a;*  —  a?* 

by  putting  x  =  z^  and  dx  =  6z^dz^  which  will  give 

1=  2z'  +  Bz'  +  63  -f  6  log  (s  - 1)  +  C ; 
which,  since  z  =  x",  is  easily  reduced  to 

r  /^^      =  2x/x  +  3^^  4-  ei^x  +  log  (.i^*-  1)  +  C. 

J    X^  —  TT 

/nx      _j_  ^ 
— Y'  dxj  may  be  freed  from  the  nega- 
x^  +  X'  -^ 

tive  index  of  x  in  its  numerator,  by  multiplying  its  numer- 


FEACTIONAL   FORMS.  363 

ator  and  denominator  by  a?*,  wliicli  reduces  it  to 

rax~^  +  1  ,  r  a  +  x^   J 

J  ~l ^-  ^^  =  /  -^ 1  ^-^5 

•^    a?*  +  a; '  =^  *^  x^  +  x^ 

wliicli,  by  putting  x  —  s^^,  becomes 

12  r^l,  ."&  =  12  f^  +  12  f^ 
=  a  [62^  -  125  +  12  log  {z  +  1)] 

~T~~ — I '  ^y  putting  a?  =  2^°,  becomes 
a;^/+  a?' 

J  ^  +  z^  J  z^  +  1  J  \  z^  +  1} 

=  2^-5.^  +  10/^-^^3; 

noticing  that  this  integral  can  be  easily  found  by  diverging 
series. 

(9.)  If  the  surds  which  enter  into  the  differential  coeffi- 
cients of  a  given  binomial  form,  contain  the  simple  power 
of  the  variable,  then  it  is  clear  that  the  differential  may  be 
rationalized  in  like  manner  as  before. 

Thus  the  differential  [3  {a  +  hxf  +  2  («  +  hxf]  dx  is 
rationalized  by  putting  a  +  hx  =  z^,  which  gives 

6Mz 

ax  =  — - —  ; 

and  thence  the  proposed  differential  becomes 
(33V  2^)  x^^^ 


364  FRACTIONAL  FORMS. 

which  is  of  a  rational  form,  which  reduces  the  integral  of 
the  proposed  differential  to 

the  same  result  that  the  immediate  integration  of  the  pro- 
posed differential  will  give. 

Ai       ^x.     '  ^        1      ^   («  +  hx)^  +  (a  +  hxf  ,      .  ., 

Also  the  mteffral   of  ^ H ~  ax,  is  easily 

{a-\-hxf  ^ 

rationalized  by  putting  a  +  2>a7  =  s^^,  which  gives 

Vlz'Hz 

ax  =  — Y —  ; 

and  thence  the  proposed  differential  is  reduced  to  the  rational 
differential 

122^^       1i2z''dz 


whose  integral  is 

4z''        1 23"                4  (a  +  hxf^       1 2  (a  +  hxf^       ^ 
~bh   +  136"  +  ^-  U  +  135 +  ^' 

adx 
The  integral  of  — -t-y^ — j-r   is   rationalized  by  putting 

2^dz 
a^  —  hx  =  s^,   which   gives  dx= ^^j—  ;  and  thence  the 

proposed  differential  is  reduced  to  the  rational  differential 

/adz     _^  r   dz  1  r   dz 

z^  —  a^  ~  2J  z~-^  ~~  2J  z~+a ' 

whose  integral  is 

log4/iZ«  +  c,     or      fJ.^-l-^  =  logCC-Zl^t 

^  ^    z  +  a         '  J  z^  —  a^  ^      \z  +  aj' 

as  required. 


RATIONALIZING   INTEGRALS.  365 

icdis 
The    integral   of        '    - — r    is    rationalized    by    putting 

1  -f-  a?  =  s^,  which  gives  dx  =  ^zdz  and  x  ^=-  z^  —  1,  which 
reduces  the  proposed  differential  to  2  {f  —  1)  dz^  whose  inte- 
gral is 

xdij 
The  differential ^ — -^ ,  by  putting  1  +  a?  =  s^,  is  re- 

(1  +  a?)^ 

duced  to  the  rational  differential  Zz^dz  —  Szdz,  whose  in- 
tegrai  is  -^ ^ — [-  kj,  as  required. 

(10.)  We  now  propose  to  show  how  to  rationalize  differ- 
entials whose  coefficients  involve  the  square  root  of  an  ex- 
pression of  the  form  a  -{-  hx  +  cx^,  or  an  expression  that 
may  be  supposed  to  be  comprehended  by  this  form  or  come 

under  it. 

dx 
Thus,  to  rationalize  the  differential  '  ,         ^  ,^  we 

Va  -\-  bx  +  cx^ 

assume 

a  +  hx  -{-  cx^  =  (x  -{-  zfc  =  x^c  +  2xzg  +  z% 

which  gives     a  -{-Ix  =  2xzg  -\-  z^c,     and  gives 


j/^ 


+  bx  -\-  cx^  Va  +  hx  + 

a?  =  - 


\/G 

and  thence  x  = ^  ; 

2gz  —  0- 

by  adding  z  to  a?,  we  have 

{2gz  —  h) 
and  by  taking  the  differential  of  the  value  of  x  we  also  have 

_  —'^o{a  —  bz  +  ez^)dB 
'^'^  -  {2gz  -  hf  • 


366  RATIONALIZING   FORMS. 

Hence,  from  the  substitution  of  these  values  in  the  given 
differential,  it  becomes 

2  {a—  hz  -h  C2^)  i^cdz  ^ 
"  {2cz-h)  ia-hz  +  cz-y 

or  by  reduction  we  have  the  differential  -^ —     ,    ,  which  is 

reduced  to  —  -r-; jr  --■    yc,  which,  by  integration,  gives 

2 — \- z  for  the  mteerral  /  — -, ^ ^ ,  as  re- 

quired. 

If  c  is  negative,  or  the  proposed  differential  of  the  form 

dx 
.        ,        ^= ,  we  may  find  the  factors  of  a  ■\-hx—  ex-  by 

r  (I  -j-  OX  —  CXi 

solving  the  quadratic   equation   a  -\-l}x  --  cx'  =.  0,  or  its 

equivalent  x^ x  =  -]  whose  roots  will  be  found  to  be 

c  c 

X-  ^^  and     X-  2^ 

the  first  being  positive,  and  the  second  negative  when  a  is 
positive. 

Hence,  if  a'  and  V  stand  for  the  first  and  second  of  these 
roots,  we  shall  evidently,  from  well-known  principles,  have 

CL  OX 

{a!  —  x)  (x  —  h')  equal  to  — h x^\  consequently,  the 

G  0 

dx 


proposed  differential  is  reduced  to  — = 

V{a'  -x){x-b') 

To  rationalize  this  differential,  we  may  assume 
(a'  —  x)  (x  —  h')  =  (x  —  h'fz"     or     a'  —  x  =  (x  —  Z>')  z', 

,  .  ,      .  a'  +  h'z' 

which  gives  x  ~  — ^  ; 


RATIONALIZIJS^G    FORMS.  867 

wKose  differential  gives 

_2(h'  —  a')  zdz 
"^'^  -        {z'  +  If       ' 

Hence,  since    Via'  —  x)  {x  —  V)  =  l-r-r^)  ^> 

dx  _  2(73 

^^  ^^"y  S*^*  VeV{a'-^){x-b')  =  (?  +  "r)'  '"'""''  '' 
a  rational  differential,  since  z  is  not  affected  by  the  surd 
sign. 

Remark. — If  tlie  proposed  differential  is  of  the  form 

j^{a  -\-  Ix  +  car)  dx, 

bj  multiplying  and  dividing  hj  \/{a  +  hx  i-  car)  we  have 

{a  +  bx  -{-  cify)  dx 
\/{a  +  hx  -{-  cx^)  ' 
wbicli  is  equivalent  to 

adx  hxdx  cx'^dx 

+   -77—1 : ?:  + 


|/(a  -{-hx  -\-  cx^)        \/{a  +  hx  +  car)        \/{a  +  bx  +  cx^y 

in  wbicb,  as  in  the  preceding  examples,  the  irrationality  is 
brought  into  the  denominator  of  a  fraction ;  which  we  may 
clearly  always  suppose  to  be  done  in  practice. 

To  illustrate   what  has  been  done,  take  the   following 

EXAMPLES. 

dx 
1.  To  find  the  integral  of  the  differential  — — - — - . 

Va  -\-  Gx^ 
Because  the  iirst  of  the  preceding  general  forms,  or  the 
general  form  in  (10),  by  putting  J  =  0,  is  reduced  to  the  pro- 
posed example ;  it  clearly  follows,  that  by  putting  Z>  =  0  in 
the  results  in  (10),  we  shall  get  the  corresponding  results  in 


368  RATIONALIZING  FORMS. 

the  preceding  example.     Hence,  we  shall  get  -—  log  Cz  for 

the  integral^  .— --^>  as  required. 

dx 
2.  To  integrate  the  differential 


a  -\-  hx  —  u^' 


By  putting  1  for  c  in  the  second  of  the  preceding  general 

forms  (see  p.  366),  we  shall  have  x  =  — ^ r^  ,  a'  and  ¥ 

being  the  roots  of  the  equation  a^  —  hx  —  a=0;  and 

/v(a/L-a^)  =  -^*'^""'^  +  ^' 
or  since  (from  p.  366),  z  =  y  - — j-, ,  we  shall  have 


/dx  _  p       o        -1    /^' 

i/(a  4-hx  —  x^)~  '^  '   X  - 


—  X 
\/{a -^-hx  —  x^)  ~  ^       *^  ''""      ^  X  —  b'' 


dx 
3.  To  integrate  the  differential  — -j-^ — ^~¥.  • 

It  is  manifest  from  the  nature  of  a  differential,  that  the 

dx 
integral   of  the   differential  —  in  3,  must  be  ex- 

Va^  +  (rx" 
pressed  by  a  logarithm,  and  be  of  the  form 

divided  by  <?,  or  of  the  form  log  [ex  +  |/(a-  +  crx')]. 

dcd'ie 
Since  the  differential  of  this  is  cdx  -\ ,  ...  ' '  \,  „, ,  which 

divided  by  <?,  is  easily  reducible  to 

dx 


j^{a?  +  crxr) 


j-r-  X  {ex  +  Va?  -\-  6'V), 


KATIONALIZING   FORMS.  369 


wliich,  divided  by  the  quantity  ex  +  \'a-  +  c-V',  gives  tlie 
proposed  differential,  as  it  ought  to  do.  Consequently,  after 
the  addition  of  a  constant  to  the  preceding  integral,  it  will 
represent  the  complete  integral  of  the  proposed  differential, 
as  required. 

4.  To  find  the  inteo^ral  of  the  differential  --7-^ — -^rr.- 

^  X  1^'  {a?  -j-  6' V) 

Proceeding  as  in  the  last  example,  we  have 

dx  _  dx 

X  \/{a^  -\-  cV)        x^ 

or  putting  -  =  y^  we  get 


V  /(a-  „\ 


dx  dy 


or  from  3,  we  have 

r        dx  I  l/fl^  +  c'x'  +  a 

as  required.     And  thence 

/djf.  1  

^7(^-F^^)  =  -  ^°S  -  log  0  V{a^f  +  e^)  +  «y. 

5.  To  find  the  integral  of  -yfr-ZTT^  ^^  rationalizing  it. 

It  is  manifest  that,  as  in  the  Diophantine  Analysis,  we 
may  assume 

l-x'  =  {l-xzf  =  l-  2x2  +  a;V, 

22 

and  thence  get  x  =  j— ^ ,  which  gives 
,  2{1-  2')d2  ,        ,  ,-,  „,        ,  1-2* 


870  RATIONALIZING  FORMS 

Hence  we  shall  get 

^  4/(1  -  »=)        V  1  +  ^ 

Another  form  of  the  integral  of  the  proposed  differential  is 
well  known  to  be  expressed  by 


/dx  .      1         ^ 

-771 ST  =  sm-^ic  +  C. 
4/(1  —  a?-) 


Since  s  = ,  it  is  clear  from  tan  2^  = 


1-3-' 

that  we   shall  have  tan  2^  =  —t=L-z=z^:   consequently,  we 

Vl-x' 

shall  have  2  tan-^;^  equal  to  sin-^a?,  which  is  as  it  ought 
to  be. 

dx 
The  integral  of  Tr^j~^^=^2  is  found  by  putting 

2h 

4:hzdz 


V  2hx  —  x^  =  xz^     or     Ihx  —  ar  =  a?-2l     or     t^^— s  =  a;, 

I  +  z^  ' 


and  thence  c?u7  _ 

„  .  Ihz 

Hence,  since      xz  — ^ ,    we  get 

r    "^  r  "idz      ^    ^ 

as  required. 

Finally,  the  integrals  of  — j- — '■ and  ^ 


x\/{\  +  X  +  X')  ^/{2ax  +  x^) 

can  be  done  in  like  manner  to  4  and  3. 


INTEGRALS.  371 

dx 

6.  To  find  the  integral  of  the  differential '■ — = . 

' r    =    — rr    -f    C   (C    =    COnSt)  I 

{l  +  aPf        Vl-hx'  ^  ^ 

dx 

since  its  differential  is 1  the  proposed  differential. 

(1  +  xy 

Ke MARKS. — It  is  easj  to  perceive  that  the  differentials 

dx  ,  dx 

and 


j^{a  +  bx  -{-  car)  4/  {a  -\-  bx  —  cx^) 

(given  at  pp.  %Qb  and  366)  admit  of  the  following  useful 
transformations. 

It  is  clear  that  we  shall  have 

dx  dy 


and 


\/{a^hx-\-  cx^)        \/c  4/  (A-  4-  y^) 
dx  c?s 


4/ (a  +  6»  —  cx'^)  ~~  i/c  VB^  —  z'' 
It  is  clear  that  in  this  way  the  integrals 

/dx  A    r       ^^ 

4/  (1  +  a?  +  X-)     ^^^       J    4/  (1  +  a?  —  x^) 

are  reducible  to  the  forms 

r*  dx  ,         /*  dx 

which,  by  putting  a?  +  ^  =  s,   and   x  ~  ~  =z  z\   give 


'^^  lo.,cf.  +  |/(.^+|) 


3        .\ 


'o 


872  INTEGRALS. 

and    -—    /  — --E =  —^  arc  (rad  =  ~  and  sin  =  z\ 

Those  correspond  to  the  right  members  of  the  first  and 
second  equations  reckoned  downward ;  which  are  made 
integrable  bj  putting 

X  +  -^  =  2     and    X  —  -  =  z', 
as  below. 

(11.)  Supposing  the  variable  enters  the  coefficient  of  dx^ 
in  the  form  a?"*  {a  +  hx^ydx,  called  a  binomial  differential^ 
such  that  the  exponents  ?»,  ?i,  and  ^  are  integral  or  frac- 
tional, positive  or  negative ;  then  we  may  clearly  proceed  to 
simplify  its  integral  as  follows : 

1.  It  is  manifest,  that  m  and  n  may  always  be  regarded 
as  being  integers,  since  they  may  always  be  reduced  to 
integers  by  introducing  a  new  variable. 

2.  n  may  be  supposed  to  be  always  positive ;  for,  by  put- 

1  1  .  . 

ting  -  for  a?,  a?"  becomes  —  =  ?/-",   in  which,  when  n  is 

negative,  —  n  must,  of  course,  be  positive,  and  in  y-""  the 
exponent  is  positive,  as  proposed. 

3.  Hence,  the  integral  I  x"^  {a  -{-  Ix'ydx  may  always  be 

supposed  to  be  of  the  same  general  form,  in  which  m  and  n 
are  integers  rnd  n  positive,  while  m  and  <p  may  be  nega- 
tive, and  at  the  same  time  p  may  be  fractional. 

4.  Resuming  the  proposed  differential,  without  regarding 
the  preceding  reductions,  and  putting  a  -f-  hx^  —  2,  we  shall 


INTEGKALS.  873 


get    X  =  I — T—ji   wliicli  gives 


1  i-i 

dx  =  — I  {z  —  a)  "     dz  ; 

consequently,  the  integral  of  tlie   proposed  differential  is 
reduced  to  that  of  the  inteorral 


■Q' 


a)  ^       zHz (1); 


which  clearly  shows  that  its  integral  can  he  found  when 

— is  an  integer 'j   this  heing  called  the  condition  of  in- 

tegrdbility  of  the  expression. 

Because  /  x^  (a  +  hx'^ydx  is  equivalent  to 

Cx'^  +  '^P  {ax-''-{-hydx, 

if  in  this  we  put  ax-""  -\-  h  —  z  and  proceed  as  before,  the 
integral  reduces  to 

m  +  1  /» 

- q^-y (.- J)- ^^^-' ..<;.....  (2). 

Hence,  the  integral  can  clearly  be  found,  when  the  con 

dition  of  integrdbility  is  represented  by  — f-  i?  =  an 

integer. 

EXAMPLES. 


J  x{a  +  bxy 


1.  To  find  the  integral  of    I  x  {a  -{-  bx^^dx. 

Here,  since  x"^  is  represented  by  x,  we  have  7n=  I,  and  as 
hx^  is  expressed  by  Ja?^,  we  have  ^i  =^  2 ;  consequently,  the 


874  INTEGRALS, 

condition  of  integrability  becomes =  — ^ —  =  1  is 

satisfied,  and  the  integral  must  be  exact. 

By  substituting  the  values  of  vi^  n,  and  p,  we  shall  have 

fx  {a  +  hx^)  dx  =  ~.  f{2  -  af  z'dz 

1         2    i 
=  [since  (s  —  a)°  =  1]  ---  x  ^  2''  +  C 

2.  To  find  the  integral^  a?--  (a  +  lic'Y^dx. 
Since  w  =  —  2  and  ti  =  2,  we  have 

m  + 1         .  11 

'     — ^-  4-^=----=-!  =  an  mteger, 

which  satisfies  the  second  condition  of  integrability.     Hence, 
we  shall  have 

fx-''  {a  +  Ix-y^dx  ^fx-^  (««-'  +  h)~^dx  = 

3.  To  find 

/  x^  {a  +  hx^f  dx  =  ^2  /  {z  —  a)  ^  dz 

/dx 
(1  +  x'^)' 
Since  the  integral  is  equivalent  to  /  (1  +  x^)~^dx^  vrc  have 


INTEGRALS.  375 

m  =  0, =  ^ ,  and  as  />  =  —  ^  we  have  -  —  -  =  —  1 

an  integer,  and  we  have 


z 


'*  +  C  = 


^  +  ^- 


4/(1 

/x'dx 

Because  the  integral  is  equivalent  to    I  af  {oc^  +  a^)~^  dx^ 
we  have        m  =  5,    /i  =  2,    and  j9  =  —  1, 

and  =r  -  =  3  an  integer, 

7i  2 

and  we  have 

fa^  {x''  +  a')-'  dx  =  ^  f{z  -  ajz-^dz  = 

-  -  a-^  +  -  log  s  +  C  =  -^ ^  +  —  log  (aHiC')  +  C  ; 

which  may  also  be  found  by  converting  the  fraction  -^ 5 

into  a  series,  arranged  according  to  the  descending  powers  of 
X,  and  then  taking  the  integral  of  the  quotient 

Since    m  =  5,    71  =  2,    and   j?  =  —-^    the  equation 

/I        r  "l±l_i 

x""  (a  +  hxydx  =  — „^^  J  iz  —  a)  »      z^dz 

becomes 

fa^  {a  +  Wy^dx  =  ^  f{z  -  afz-^dz  = 


376  INTEGRALa 

7.  To  find  the  integral   /  ar^  (a  -f  hjr^fdx. 

This  integral  can  clearly  be  easily  found,  since 
m+l        —1+1 


0, 

1  ^  m  +1     1 

and 


^dz 


1  /»  m  +1_^ 

is  reduced  to       7:  I  {z  —  a)-^z^dz  =  7:  I 

SJ  ^  ^  S*/  z  —  a 

By  putting  s  =  y^  we  have 

dz  =  8?/^cZy,    and    s*c?s  =  3yWy, 

and  thence  we  have 

By  putting  y  =  va^,  the  integral  /  -  is  reduced  to 

/  -3 -■ ;  whose  integral  can  be  found  from  the  principles 

at  page  371,  «fec. 

8.  To  find  the  integral  /  x~'^{a  -\-  hx^)  dx. 

/hx^ 
x~'^  (a  +  hx'^)  dx  equals  a  log  aj  +  —  -f-  C,  as  re- 
quired. 


SECTION  IV. 

EEDUCTIONS    OF   BINOMIAL    DIFFERENTIALS    TO    OTHERS    OF 
MORE   SIMPLE   FORMS. 

(1.)  These  reductions  generally  result  from  tlie  differen- 
tial dxy  =  ydx  +  xdy^  wliich  gives 

ydx  =  dxy  —  xdy    and      /  ydx  =  xy  —  I  xdy^ 
or  /  xdy  :=  xy  —  I  ydx, 

whicli  is  called  integyvition  hy  parts ;  and  reduces  the  in- 
tegral /  ydx  to  the  integral  /  xdy^  or  the  integral  /  xdy 
to  /  ydx. 

Thus,  if  we  represent  {a  +  hx'^y  by  2^,  we  shall  have 
{a  +  hx^'Y  =  zP,   and  thence 


{a  +  hx^'Yx'^dx  =  z^d  I  x^ 


dx  =  z^d 


m  +  V 
which  gives 

fx'^zHx  =  zP  — ^^ ^^~  fx"^  +  ''zP-'^dx ....(«); 

J  m  +  1       m+lJ  ^  ^' 

which    reduces    the    integral    /   x"^z^dx    to    the    integral 

/  g,m  +  n^p-i^^^  jjj  which  p  is  diminished  by  unity,  while  7fi 

is  increased  by  71. 

Also,  from   x'^z^dx  =  x^'-^'^^d  l{a  +  hx^^yx^'-^dx, 


378  INTEGRATING. 

we  shall,  as  before,  get 

wliich  shows  that  the  integral  /  x'^z^dx  is  reduced  to  the 
integi-al  jx^-''z^  +  ^dx. 

Because  the  integrals  in  the  right  members  of  (a)  and  {h) 
admit  of  like  changes,  it  clearly  follows,  if  j9  is  a  positive 
integer  greater  than  1,  while  m  +  1,  w  +  n  +  l,  m-}-2;i-f  1, 
&c.,  are  finite,  that  the  exponent />  will  finally,  by  successive 
applications  of  (a),  be  reduced  to  unity,  and  thence  the  in- 
tegral I  x"*  {a  +  hx'ydx  will  be  determined ;   and   in   like 

manner,  from  (^),  if  jt?  is  a  negativ^e  integer  numerically 
greater  than  1,  while  h,  p  -\-  1,  p  -\-  2,  &;c.,  are  finite,  it  is 
manifest  that  the  integral  will  be  reduced,  by  successive 
applications  of  (5),  to  an  integral  in  which  a  +  hx''  will  enter 

in  the  form  (a  +  5a?") -^  = ^— ;  consequently,  agreeably 

to  what  has  heretofore  been  shown,  the  integral  will  be  re- 
duced to  the  integral  of  a  fractional  differential,  having  a 
rational  denominator,  and  is  to  be,  according  to  what  has 
been  shown,  regarded  as  known. 

(2.)  From  z  =  a  +  Ja?",  we  get  a  =  z  —  hx''  and 
h  =  ^ar""  —  oa?"",   and  thence 

afx^'z^dx  =  J  x'^'zf'^^dx  —  ifx^^^^'z^dx, 

and       ifxTz^dx  =  J  x'^-^'z^  +  \Jx  —  ajx^-'^z^dx. 
Since,  by  putting  p  +  1  for  p  in  (a),  it  reduces 

faTz^^'dx    to    gp^i-^!!^  __iP  +  1)  ^^-^  fx^^-z^dx, 
J  m-f-1  m-\-\    J 


J  aim-^l^  a  (m  -hi)         J 


INTEGRATmG.  879 

tlie  first  of  the  preceding  equations  gives 

a.m+1  {pn-^n  4-  m.  +  1)?; 

a  (??2  +  l)  a  (m  -\-  1) 

; ("); 

and,  since  by  changing  m  into  m  —  n^  and  p  into  p  +  1,  in 
(rt),  it  gives 

/m— n  +  i  (n-\-V\nl)    r 

?7i  —  n  +  1        m  —  /i  -|- 1./ 

which  being  substituted  for  /  a?"^~"s^"^  V^^  in  the  second  of 
the  same  equations,  we  readily  get 


dx 


,P+i 


{in  —  n  -f-  1)  a 


jx'^''sP(lx...{d). 


{pn  +  m  +  1)  h  {pa  +  m  +  1)  ^  ■ 
It  will  be  perceived  that  in  (c),  the  proposed  integral  is 
reduced  to  an  integral  in  which  rn  is  changed  into  7)i  +  ^, 
while  in  (d)  it  is  changed  into  in  —  n.  It  is  also  clear  that 
integrals  which  can  not  be  reduced  by  {a)  or  {h\  or  with 
difficulty,  can  often  be  easily  reduced  by  {c)  or  {ti).     Thus, 

the  integral  I  x--  (a^  -i-  o[r)-'^dx,  in  which  m  =  —  2,  n  =  2, 

jp  =  —  1,  a=za^^  and  h  =  1,  is  by  (c)  immediately  reduced  to 

Cx-^{a/-\-x-)-^dx  zzz 

ar-2  +  i  _24-2-24-l 


/.— .; 


X    /  x-''^'-27hlx 


a^(-2  +  l)  aH-2  +  1) 

1       i    r   dx  1       1  ^     ,  aj     ^ 

a^'a?       a' J  a-  -\-  x^  a-x        a*  a 

In  like  manner,  by  {d)  the  integral   /  cc'^  (a-  +  x'^)-'^dx  is 
easily  reduced  to 

fx'  (p}  +  x')-'dx  =  '^~^  log  (a^  +  x')  +  C. 


880  INTEGRATING. 

(3.)  Multiplying  the  members  of ;?  =  r/  4-  Jjf  by  x^'z^-'^dx, 
and  taking  the  integrals  of  the  equal  products,  we  have 

J^xTz^diii}  —  aJx^'z'P-^dx  +  ljx'^'^''z^-^dx. 

To  the  products  of  the  members  of  this  by  — — - ,  adding 
the  corresponding  members  of  (a),  we  get 

\         m±V  J  m  +  l       m  +  lJ  ' 

or  its  equivalent 

fx'^z^dx  =  z^  — —- — r  +  ^!L-     [x'^'z^-^dx.  ..(e)', 

which    reduces   the   integral      /  x^z^dx     to    the    integral 

/  x'^z^'^dx,  in  which  z^  is  changed  into  2^"^  Thus,  if 
2  =  a^  +  a?^  we  have  j9  =  1,  a  =  ar,  and  n  =  2,  and  thence  get 

a?"*2^a?  =  z ~  H —  /  x'^dx 

m  +  3       m-\-^J 

^  '^  7/z  +  3       m  4-  3  m  +  1  ' 

which  is  clearly  the  same  result,  that  the  immediate  integral 
of  the  proposed  differential  will  give.  If  ^  stands  for  a 
positive  integer,  it  is  clear  that  successive  applications  of  the 

above  formula  will  reduce    /  a?'"  z^  dx  to 

Jx^'z^-^d.e,  fx'^z^-^dx fx'^dx. 

Changing^  in  (c)  into  2>  +  1?  multiplying  its  members  by 
(/>  +  l)7i  +  m+l,  transposing,  &c.,  we  have 


^'^'  -7^—7-.-  + 


INTEGEATING.  381 

wliich  reduces 

ix'^z^dx     to    fx'^zP+'^dXj 

and  Cx'^'z^^hlx    to    fx^'z^+'^dx, 

and  so  on.     Thus,  if  jp  =  —  8  we  have 

y  .'".-c^^ ..  — ^_-_y  .'^.-.z.; 

and  then 

x'^z-^dx  = — ^—  /  x^z-^dx. 

7ia  na         J 

Hence,  if  z=za^  -\-  a?  and  7n  is  a  positive  integer,  the 
integral 

J  X"  -[-  or 

x^ 
which,  by  converting    ,,      — ^  into  a  series  arranged  accord- 
ing to  the  descending  powers  of  x^  can  now  be  easily  fonnd 
by  the  common  methods  of  integration. 

We  will  now,  for  convenience  in  what  is  to  folk>w,  collect 
the  preceding  formulas  into  a 

(4.)  Table  of  Formulas  for  the  Redtiction  of  the  Litegral 

Cx'^ia  +  hxydx  =  fx"'zHx, 
I. 

fx'^zHx  :=  Z^-—^   -  ^^~    fx'^^^zP-^dx. 

J  m  +  1        m  ■\-  \  J 


382  INTEGRATING. 


4 

II. 


III. 


•^  aim  +  1)  « (m  +1)       J 


a{fa  +  1)  a  {m  +  1) 

IV. 


V. 


futTz^dx  =  ;sP ?^^^^ + ^^ — -,  fx^'z^-'dx. 

*>  vn  +  m  +  1      jm  +  m  +  I*' 


jpn  +  m  +  1      2^n 
VI. 

^  a(^  +  l)?i  a(^H-l)n      J 

This  table,  under  a  different  arrangement,  is  substantially 
the  same  as  that  of  Mr.  Young,  at  page  42  of  his  "  Integral 
Calculus ;"  noticing,  that  our  formula  I.  takes  the  place  of 
his  formula  II.,  which  is  incorrect. 

(5.)  To  perceive  the  use  of  the  formulas,  take  the  follow- 
ing 

EXAMPLES.  * 
1.  To  find  the  integral  Jx-^{l—x'fdx, 

Q 

Since  m  =  —  4,  n  =  2,  ^  =  - ,  a  =  1,  and  J  =  —  1,  it  is 
clear  that  formula  I.  reduces  the  integral  to 


INTEGRATING.  383 

fx-'z^dx  =  Z^^~fx-'^'3^~'dx 

and  another  application  of  L,  reduces    I  x~^  (1  —  x^ydx    to 

/-.  oA^         C       dx  4/(1  —  07^)  .       ,  ^ 

^  ^   X      J   \/{^  —  x-)  X 

hence,  we  have 

fx-'  (1  -  x^fdx  =  -  ^,^'  +  S^-^'  +  sin-^aj  +  C. 
•/  ^  3,r^  a? 

2.  To  find  the  integral  fx^  (1  —  .'»')-^(/.z;  ==  Jx's-hix. 

Since  m  rz:  5,  n  =  2,  ^  =  —  3,  a  =  1,  and   Z>  ==  —  1,  from 
II.,  we  have 

/  x^2~^dx=  — J /  xh~^dx; 

and  another  application  of  ii.  reduces 

/  x^2~'dx     to     /  x^2--dx  =  — ~ /  X2~^dx. 

Hence,      /  x^2-^dx  =  —^ ~  +  /  xz-^dx' 

since     J  xz'Hx  =^j  J^J^^.  =  -  ^  ^^S  (1  -  ^)  +  C, 

the  integral  becomes 

/x^dx      _  a?*  x^  log  (1  —  a?^)       ^ 

(1  -  x'Y  ~"  4  (1  -  £C^  ~  2  (1  -  ar)  2         "  "^     * 

3.  To  find  the  integral  Jx-^{a^  +  a^2)-icZ;r. 

From  m  =  —  4,  m.  =  2,  ^  =  —  1,  a  =  a-,  and  J  =  1,  we 
get,  from  iii., 


384  INTEGRATING. 

fx-'  {a?  +  x^)-\lx  =fx-'3-'dx  =  ^^^  -  \fx-h-hix 

and  from  another  application  of  ill.,  we  have 

/x-'^z-\ix  =  —  —^ /  ~^z-^dx  = 7.—  I  -Trrv-J,' 

Hence,  we  have 

and  since 

dx 

\_CJ^__\  /*IZI-ltan-i-- 

aVa^  +  ar^~a^'      J   ^      ^  ~  a^  a' 

a" 

consequently,  we  shall  have 

4.  To  find  the  integral  J  x^  {a^  +  x^)-^dx. 

From  m  =  5, 71  =  2,  ^  =  —  1,  a  =  a-,  and  J  =  1,  we  shall, 
from  IV.,  get 

fa^iw"  +  aP)-'dx  =fx'2-hlx 

x^      aV      <x* ,       ,  ^        „,       ^ 

5.  To  find  the  integral    /  x^  {a-  +  x-f\lx. 

From  ??7  =  4,  ??  r=  2,  ^^  =  --,  a  =  «-,  and  J  :=  1,  v.  gives 


INTEGRATING.  385 

Jx"^  (a'  +  x^ydx  —Jx^^'dx  =  zp%  +  ~  J  x^2~'dx, 
Froitf  IV.,  we  reduce 

/  x^sT^dx     to     x^z~^dx  =  ^^  %; rj  ^"^~^dx, 

and  another  application  of  the  same  formula  reduces 

Jx-z~^dx     to     Jx"2~^dx  =  2^^  —  ^-J  z~^dx; 

consequently,  since 

fz-Ux=  f ——  =  log  [X  +  4/(a^  +  x')]  +  log  C 

J  ^  {a^  +  xy 

=.logC[.+  ^(.^  +  .=)]=logf-L4!+^), 
by  putting  C  =  - .     Hence,  we  get 

I  x^  {cr  +  x-ydx  =  — —^ —  4- 

for  the  sought  integral,  supposed  to  commence  with  x. 


dx. 


6.  To  find  the  integral  Jx{a}  —  x")   ''dx  =J 

Since  7/2  =  1,   ?i  =  2,  jp  =  —  -  ,   a^=^  ar^   and  5  =  —  1, 

It 

we  get,  from  vi., 

C     -a,  S~V         1      f      _i, 

/  xz   ''dx  =  — s ^  ]  xz    '^dx ; 

consequently,  since 

we  shall  have 
n 


886  INTEGRATING. 


in  which  C  =  the  constant . 

a 

7.  To  find  the  integral  fx""  (1  —  ar)~^dx. 

From  m^=n,  71  =  2,  _p  =  —  ^ j  a  =  1,  and  h=  —1,  we 
readily  get,  from  iv., 

fx"'{\-x')-^dx  = 

vi  m     J 

Substituting  successively  the  odd  integers,  1,  3,  5,  &;c.,  for 
w,  in  this,  we  have 

r      x'dx       _        a; V  (1  —  a^')        2  C      xdx 
J  Vi\-^)  ~  3"  "^  3^  1/(1  -a^)'        ' 

from  which  we  readily  obtain  Mr.  Young's  results  at  p.  44 
of  his  "Integral  Calculus."  And  putting  the  even  integers, 
0,  2,  4,  6,  (Sec,  successively  for  la  in  the  preceding  formula, 
we  in  like  manner  get 

r       dx  •      1      ,   n 

r     afdx _  _  a?|/(l  —str)        1  /* dx 

J  |/(l-(r)  ~  2  "^  2^  4/(1  -  xy 

and  so  on ;  from  which  Mr.  Young's  results,  at  p.  45  of  his 
work,  can  easily  be  found. 

8.  To  find  the  integral  y*ar-"»(l-a;^~W 


I2^TEGRATING. 

887 

Since 

m  = 

—  7n 

,   n 

=  2,p 

1 

"~       2' 

a 

-1, 

and  h  = 

-1, 

we  have, 

from 

IIL, 

fx-^'il- 

-  xy^dx 

m  —  1  m  —  w 


\/  (1  —  a;^)  m  —  2  /*  c?^c 


-  + 


'.  -  iJ  x'''-\ 


(m  —  1)  x"'-^       in  —  U  £c"*-y  (1  —  aj-) 

If  we  put  3,  5,  7,  &c.,  for  m  in  this  formula,  we  have 

r dx  _  —  1 1  f__dx_ 

J  a?V(l-^  ""  ^1  _  ^^       ^-^  «?(1  -  x'f 

r  dx  _        |/'(1— .'?-)        3/*         dx 

and  so  on;  and  putting  0,  2,  4,  &c.,  for  m,  we  have 


2\  » 


=  sin~^^ 


r     dx      _  _  |/(i-^^) 

r        r7a?  _  _  4/(1 -.-z-^)        2  T ^ 


and  so  on,  to  any  extent.  The  preceding  results  agree  witli 
those  of" Mr.  Young  at  pp.  46,  47,  and  48  of  his  "Integral 
Calculus  ;"  noticing,  that  the  integral 

J  x^i^-x')  °  X  ^^ 

can  not  be  found  by  the  preceding  process* 


58  INTEGRATING. 

9.  To  find  the  inteerral 


Here  m  =  m—  - ,  ri  =  1,  a  =  2a,  5  =  —  1,  and  /?=—-, 
and  theiice,  from  iv., 

f 


C2a-a?)ia?"^-i       (2??i-l)a  /*    ^^-^c^ 


x^dx 


V2cix—xF 
x'^-^V^ax-n?       (2m-  -l)a  f    x^-^dx 


+ 


/: 


m  m        J   y2ax  —  x^\ 

which  clearly  shows  that  bj  a  sufficient  number  of  repe- 
titions of  the  process,  the  integral  will  be  reduced  to 

/dx  .     ,  X       ^ 

, =  =  versin-^  — f-  C. 
V2ax  —  3r  « 

10.  To  find  the  integral  Jx"^  (a^  +  x-)-^dx. 

Since  ?i  =:  2,  az=i  a^^  and  J  =  1,  we  get,  from  vi., 

/  x^'z-Hx 
l^rt^  +  ^yp +  13,^  +  1      2{l-2J)  +  m+l  r 

■=. ^^ ±2-1 !___   /   r^m^y-p  +1.7^ 

2a'i^-\)  2ii'{p-l)      y^^      "^'^ 


aj'^  +  i 2p-S-m   r  ^  _^^ 

2a' {p-\){a^  +  x^y-'  +  -2'c' {p-l)J  ^  ^         "^ ' 


which,  by  putting  m  =  0,  reduces  to 


INTEGRATING   BY   PARTS.  389 

/dx       _  X  ^p  —  ^rdx^ 

whicli  agrees  witli  an  equation  previously  found. 

(6.)  It  is  easy  to  perceive  that  we  may  apply  the  method 
of  integrating  by  parts,  to  integrals  of  transcendental  forms. 

Thus,  to  find  the  integral  of  X  lo^'^xdx^  in  which  X  is  a 
function  of  a?,  and  n  denotes  the  nth  power  of  log  a?,  we  have 

/  X.dx  logo's?  =:  log'' .'2?  /  ILdx  -1(1  ^dx  X  ;i  log"-^a? -- ]; 
or,  by  putting   /  X.dx  =  X,  we  have 

J Xdxlog'x  =  log''xjXdx  —  j(?i -^ dx log^-^ajj  ; 

X, 

which,  by  putting  —  dx  =z  dX^ ,  gives 

X 

Jlog^'xXdx  =  log'xJXdx  —  nJdX.  log""^  x (1). 

If  71  is  a  positive  integer,  and 

Xdx=dX,,     ^dx  =  dX.,     ^dx^dX.,     &c., 

X  X  -J  ? 

are  exact  differentials,  it  is  clear  that  log"  -  ^  x  will  finally  be 
reduced  to  log*'  x  —  1^  and  thence  the  integral  finally  be  re- 
duced to  an  algebraic  form.     Thus, 

/x^      x^ 
x^  log  xdx  =:  log  a?  X  --  —  '^  +  0 ; 

/  a?-^ dx log^ ^  —  r ^og^ ^~J  I  ^^ ^^E ^^^1 
and 

/  x^  log  xdx  =  J-  log  X  —  -  I  x^  dx  =  'j-  log  X  —  j — j  »''  +  C, 

and  thence  we  shall  have 

/»*      .         1  1 

x^ dx log^ ^=  T  log'^ ^  —  Q^^logx  +  ^^ x^  4-  0. 


390  INTEGRATING   BY   PARTS. 

In  a  similar  way,  we  have 

^  7/1+  i     "="  rn  +  lJ  ^  ' 

x'^dx log" - ^  a?  = log"  -^x /  x""  log"- ^xdx, 

^  m  +  1     °  m  +  lJ  ^  ' 

and  so  on  to  any  extent,  when  ?/i  is  different  from  —  1,  or 

when  m  -f-  1  is  different  from  naught. 

Hence,  when  r/i  +  1  is  different  from  naught,  we  shall, 

from  the  requisite  substitutions  in  the  first  of  these  equations, 

/a?'" "''  ^  /  9i 

X^'dx log"  X  =:  -—^ - ^log" X  -  ——J  log" -'Xi- 

n(n  —  l),     „    „         n{n  —  l)(n  —  2),     ,    ,  ,     \  ^r^ 

If  m  +  1  =  0,  or  m  =  —  1,  we  have 

/  iC'"^??  log"a?  =   /  log"  X  =  -^ :; 1-  C. 

J  °  J    X      °  n  +  1 

We  may,  in  much  the  same  way  as  before,  find  the  integral 

/ic"*  dx         r  r  dv 

, — — =   I  x"'dxlos:-''x  =      x"'  +  ^loo:-''x  —  = 
log"  a?       J  ^  J  ^  x 

o_^ 1 #  aj'n  +  1  _  ]     - 

71  —  1  n  —  it/  a?*' 

K"*-+  ^±1  log— .+  (-;:^,J^3-^  log--.+  . . .) 
(m  +  1)"-^       Cx'^'dx  ^ 

^  1.2.8 . . .  {ii  -  ly  i^ ' 


7i— 1 


in  which  it  is  clear  that  the  integral  /  ^ can  not  be  found 


by  this  method.     If  we  put  x"^^^  ~  2,  we  shall  have 

«,  7  '^^^  1        -I  log  2! 

af"aiC  = and     log  x  =  — -^^  : 

m  +  1  °  m  +  1 ' 


INTEGRATING   BY  PARTS.  391 


and  01  course  :; = 


log  X       log  z  ' 

whose  integral  can  be  found  by  series. 

cix 
Thus,  since  d  (log  a?)  =  --- ,  we  shall  have 

X 

d  (log  x)  __      dx 
log  X  X  log  X ' 

/dx 
— i =  log  log  X  +  const. 
X  log  X 

If  (as  is  sometimes  done}  we  write  log-  x  for  log  log  a?,  log'  x 

for  log  log  log  a?,  and  so  on,  then   we  ought  evidently  to 

express  the  second,   third,   &;c.,  powers  of  log  x  by  such 

forms  as  (log  xf^  (log  xf^  &c.     Hence,  adopting  this  notation, 

we  shall  have      /  — r —  l^g^viJ  -f  const, 

J  X  log  X 

/dx 
X  log  X  rd  (log^  x)        1     ,       ,    n 
^ j3  ^               log-  a?         ./     log-  X 

dx 


f 


X  log  X  log-  a?  log^  a? 


=  log*  X  +  C, 


and  so  on.     If  we  integrate  , by  parts,  we  shall  have 

r  dz   _  _z_ r    ( J_\ _ _±_     f   dz 

J  loiT  z       los:  z       J  '^    Vloo:  zl      logr  z     J  ( 


(log  zf 


z 


lo2f  zr      J 


logs       (log  2)=^       J        (log  2)2 

-  logs  -^  Uog^y  "^  (logs/      V  ^^'^  (log  s)^ 

s      ,        z        ^       ^2z       ^     2/Sz  2.BAz       ,      , 

logs      (logs)-      (logs)-^      (logs)*  (logs)= 

s     /^         1            1.2           1.2.3  ,     \ 

=  1 1 1  +  f- "~  +  n "^  +  7t \^  +  &"•  I  const  ; 

JogsV        logs      (logs)-      (logs)'  / 


892  BEDUCTION  TO  SIMPLER   FORMS. 

/dz 
1 

in  converging  terms,  on  account  of  its  evident  divergency. 
Remarks. — 1.  Because 

/dz 
, . 

2.  If  w  =  log  z  we  shall  have  z  =  e",  e  being  the  base  of 

hyperbolic  logarithms,  which  gives  ,-- ^^  = ;    or,   since 


,3  „A 


^"  =  1  +  "+ 2  +273 +  2:3:4 +'*<=•' 
"we  shall  have 

/ifi  =  /?  +  Z'^"  + 1/"''"  +  2^3/"''^''  +  '^'^ 

=:  log w  +  1/  4-  22  +  2:3'2  +  2.3  4^  "^  "^^  +^^^s*- 
From  -w  =  log  2,  we  have  log  w  =  log  log  z  =  log-^,  &c., 
and  thence  get 


/ 


log  3  -  lor^  +  log  ^  +  — I2—  +  ^-^F-  +  <^^-  +  const. 

(See  pp.  55  and  56  of  Young's  "  Integral  Calculus.") 

(7.)  It  is  eos;y  to  perceive  that  we  may  readily  find  the  in- 
tegral of  the  differential,  af^  a^  dx^  which  involves  the  expo- 
nential a^,  in  much  the  same  way  as  before. 

«^  dx  =  x'^d , 

log  a 

we  get,  by  integrating  by  parts, 

fx^a^dx=  f^  -  ,-^  fx^-Ui-dx, 
J  log  a       log  cw 

x^-^a'dx  =  '—, ^ /  x^'-^a'^dx, 

log  a  log  a  J  ^  . 


heductio^t  to  simpler  for^is.  393 

and  so  on.     Hence,  we  shall  have 

J  loff  a-"  \  log  a 


log 


(log  af  (log  af 

noticing,  that  if  m  is  a  positive  integer,  the  last  term  within 

the  parentheses  will  be  ±     '  '    '"',7—,  accordingly  as  m  is  an 

even  or  odd  number.     It  is  easy  to  perceive  that  we  shall,  in 
the  same  way,  have 

a-^'x^'^dx 


/■ 


a-""  I  mx"'-^      7n{ni—l)x"'-^  1.2.8....m\ 

/ci^  dx        C 
— ~-  z=z  I  a^xr'^dx  is  also  easily  found  to 

be  expressed  by  the  form 

*a^  dx.  a^ 


f 


\   ^ 7?i-2    ^ {;,n-2){?a-^)^""'^ {m-2){m-S)..,.r       /"^ 

(log  g)^'-^    ra'^dx 

(m  -l){}n  —  2)...AJ     x     ' 

If  we  expand  a^  according  to  the  Exponential  Theorem 

(J),  given  at  page  51,  we  shall  have 

1,1  .   (logrt)-.c'       (looja)V 

a^  =:  1  +  log  ax  +  ^-^2       +     12   3    '^'  ^"^^ ' 

consequently,  we  shall  thence  get 

/a^  dx  _ 
X      ~" 
,  ,  ,  (logrt)^;^^  ,   (log«V^5      (los^aYx^      ,         ^ 


394  REDUCTION  TO  SIMPLER   FORAIS. 

It  is  hence  easy  to  perceive  how  we  can  find  the  integrals 

j  x'"a''tic     and      /   ~^;^j 

by  means  of  the  Exponential  Theorem;  by  converting  6^* 

into  a  series  arranged  according  to  the  ascending  powers  of  a?. 

It  is  manifest  that,  in  this  way,  we  shall  get  the  integral 

/f^  =  /|l  +  (1  +  ^oga).  +  (l  +  log.  +  ^|r)  .^  + 

»  +  (1  -f  log  a)  -  +  ^1  loga  +  -^^)  \  +  &c.  -h  const. , 

noticing,  if  a  =  ^  the  base  of  hyperbolic  logarithms,  since 
log  ^  =  1,  we  shall  have 

y  r^-^  =  ^  +  (^  + 1)  2  +  (1  + 1 "  1:2)  3-  +^^-+--^^^- 

Because  :j =  —  - — ~-  =  —  c?  [log  (1  —  a?)],  we  easily 

which,  integrated  by  parts,  by  the  successive  use  of  the  sign 
y,  gives 

zj =  —  a^log  (1  —  a?)  4-  log  a  j  a^  log  (1  —  x)  dx  ; 

and  integrating  again  by  parts,  and  so  on,  we  shall  have 

i— ,=  -«'log(l-^') 
+  log  a  a^^J  log  {I  —  X)  dx  —  (log  cifa''  j  dx  j  log  (1  —  x)  dx 

+  (log  afa^J  dxj  dxj  log  {i  —  x^dx  —  ko.  -\-  const 


REDUCTION  TO   SIMPLER  FORMS.  895 

Q  OS  C(j 

Because      —  log  (1  —  a?)  =  a?  +  -  +  o   +  T  +i  ^^-i 

the  indicated  integrations  in  the  equation  can  be  performed 
as  required  ;  and  thence  the  integral  will  be  found.  Hence 
we  shall  have 

\l        x^      x""        0     \      1         I  ^^        ^        a'*       0     \ 
<^   1(*  +  2  +  3    +  &c.)  -loga  (j^  +  -  +  3:5+&c.)  + 

If  for  a^  we  put  1  +  log  ax  4-  (log  of  ---  + ,  &c.,  in  this,  and 

perform  the  indicated  multiplication,  we  shall  easily  get  the 

value  of    /  — — ^  found  above. 
J  \  —  x 

Because  /  - — —  =  /  -z d  I  a^dx  =  /  — - — r^; da^. 

d  l—x     d  1  —  x   d  d  (1— a?)  log  a 

we  shall,  by  integrating  by  parts,  get 

r^dx^  _    3,  / 1 __1 

J  1  —  x~       \{i  —x)  log  a       (1  —  xf{\og  af 

1.2  1.2.3 


.2.3 ,    0     \   , 


{l-x)\{ogaf      {l-x)\\oga) 

(See  Lacroix  "Calcul  Integral,"  p.  93.) 
It  is  easy  to  perceive  that  the  integral 

/e^xdx    _     <i^  p 


const 


(1  +  xf      1  +  a? 

For  examples  of  integrals  of  the  forms   /  sin"*a?cos"irc?a?, 

n    -,      ^  /*sin  mx  ,      . 

(fee,  together  with  those  of  the  form  J  --g— "  dx,  &c.,  we 


396  INTEGRALS  OF   VARIOUS   FORMS. 

shall  refer  to  La  Croix  "  Calcul  Integral,"  pages  99,  &c. ; 
and  to  Young,  on  the  same  subject,  pnges  60,  &c. ;  and  for 
the  exact  Integrals  of  such  expressions,  see  Young,  pages 
71,  75. 

(8.)  1.  To  find  the  integrals 

dx  ,       C       dx 


/.  ,     — 5—    and      /  -r-r 
sm*a?cos"^iC  J  sin° 


sin"  a?  cos- a? 


Bj  putting  4  and  3  for  m.  and  n^  the  first  expression  re- 
duces to 

/dx     __  1       j_  ^  r    ^^^ 

sin*  X  cos^  x~~        3  sin^  x  cos^  x   '    SJ  sin'^  x  aos^  x  * 

and  putting  5  and  2  for  jn  and  n,  the  second  reduces  to 
dx  1  5  r       dx 


r     dx i__ 5  r 

J  sin^x  cos^a?  ~        4  sin*  a?  cos  x       iJ  i 
In  much  the  same  way,  we  have 

J  surxcos^x  smajcos-a?         J  ( 

and        / -^-3 —  =  -^-^ +  3  / 

•/  sm'^iccos^a?       sm^'r  cos  a?         J  j 


sm^  X  cos^  a? 


dx 

OS^i 

dx 


also,  from  {k)  and  (^)  these  results  become 

dx  sin  a?         1  /*  dx 


cos  a?' 


/aa?     _    sm  a?         1  /* 
cos'*  a?  ~  2  cos'^a?       2./  ( 

T  /*  c?a?  cos  07         1  /* 

and  /  -r^~  =  —  jr-"-^-  +  o  /  - 

•/  sm^a?  2  sm-a?       2^  £ 


sin  x' 

Hence,  from  the  requisite  substitutions,  we  have 

dx 


f. 


1  5  5  sin  a?       5  /*  e?a7 

8  sin^a?  cos^a?       3  sin  x  cos^a?       2  cos"  a;       2  J  cos  a? 


and 


ILLUSTRATED   BY   EXAMPLES. 
dx 


397 


/ 


sin^  X  cos^  X 


+ 


15  cos  X 


,..^-  +  -^1 


r  dx 

J  sin  a? ' 


integrals 


and 


dx 


4  sin* a?  cos  a?    '    4  sin- a?  cos  x      8  sin'^eosa?  '    8  ^  sin  a?' 
consequently,   the  proposed  integrals  are   reduced   to   the 
dx 
cos  X 
2.  To  find  the  integrals 

r  dx        r  dx  A      r     ^^ 

./  sin  a? '    J  cos  x '  J  sin  x  cos  aj 


/; 


/: 


sm  a? 


Since  cos  x  =  cos^! 


sin^  ~r ,  we  have 


•/  cos  X         I 


(ia? 


dx 

"2 


oos^  -  -  sm-  - 


cos^  - 


-J 


tan 


^  tan 


1  +  tan 


tan 


log 


1  +  tan 


tan 


+  const. 


=  log  cot  ^  (^  —  a?j  +  const. 
Hence,  since  sin  x  =  cos(^  —  ^)  ?  by  changing  x  in  the 

preceding  results  into  o  —  ^j  then,  since  the  differential  of 

this  arc  is  minus,  we  shall  have 

/dx         .       ^       X 
=  log  tan  -  +  const, 
sin  a?  ^         2 


898  INTEGRALS  OF   VARIOUS   FORMS. 

Because,  sin-  x  -f  cos-  j;  =  1,  we  shall  have 

/dx       _  /*sin^  a?  +  COST'S  ,    _  /•s.in  x  ,         /*cosaj  ,^ 
sin  a;  cos  a?     J     sin  a;  cos  a?         ~J  cos.r  J  mu  x 

=  log  sin  a;  —  log  cos  ar  -f  const  ==  log  tan  a;  +  const. 

(9.)  We  now  propose  to  show  how  to  find  the  integrals 
of  the  differentials  of  a  variable,  into  whose  differential 
coefficients  enter  exponential  and  trigonometrical  functions 
of  the  variable. 

Thus,  from 

J  A"'  sin"*  xdx  =  —  J  A"""  sin"*  -  ^  xd  cos  x, 
by  integrating  by  parts,  we  get 
/  A^sin^'ajr/aj^— A°''sin"*~^a;cosaj  +  /  cosa'c?(A''^sin'"-'a!) 

=  —  A*^^  sin  "*  ~ ^  a?  cos  x  -{-  a  log  A   /  A"^  sin"*  ~  ^x  cos  xdx  + 

(m  —  1)  /  A""^ sin"* --x  {i  —  sin^ x)  dx, 
by  putting  1  —  sin-  x  for  cos^  x.     From 

/'a- sin"*-^  X  cos  xdx  =    fk.--  dism^'x) 
by  integrating  by  parts,  we  have 

A""^  sin"* -^aj  cos  a;/7a;  — ^ ^ _-P__  /  A°^sin"*a'<:/r. 

7)%  m     J 

Hence,  from  the  substitution  of  this  value  and  an  obvious 

reduction,  we  get 

/  A"'"'  sin"*  xdx  — 

—  A'"'  sin"*  -  ^  a^  cos  a;  +  ^-^^ —  A*""^  sin'"  x  — 


(alogA)y^^^^.^^  ^^  ^  ^^  _  ^^  y^, 
{m  —  t)  /  A^^'  sin"*  a%?a; ; 


ILLUSTRATED    BY    EXAMPLES.  899 

or  transposing  so  as  to  unite  like  integrals,  we  Lave 
J  A'''' s\\r  xdx  +  {in  —  1)  J  A'''' sm"\xdx  + 


(_ 


7/1 


(a  log  A  sin  a?  —  m  cos  .t)  + 


(/;i  —  1 )  /  A^^  sin"'  -  2  xdx ; 
from  which  we  get 

/A""^  sin"' -  ^  a?    /    -,       a    • 
A'^  sm""  xdx  =  ,— i T— 5 :,  (a  log  A  sm  a?  —  m  cos  £c)  + 
(a  log  A)-  +  m-  ^      ° 

-r-,       ,  ,0     "-1,  /  A-'^sin^'-'-^ictZic (a). 

(a  log  A)^  4-  m^J  ^  ' 

In  like  manner,  we  have 

r  Ka^  m        7  A«^  COS"' "  ^75        ,      ,  ,  .  . 

/  A"""  cos"*  xdx  =  7 — -. T— ^ n  {a  losr  A  cos  x  +  m  sm  x)  -f 

J  {a  log  A)-^  +  m"  ^      ^  ^ 

^J^^l^CA-^cos—'-xdx (h). 

{a  log  A)^  +  m-d  ^  ^ 

If  m  is  a  positive  integer  greater  than  2,  it  clearly  follow^s 
that  (a)  and  {h)  will  reduce  the  proposed  integral  to  others 
in  which  the  indices  of  sin  x  and  cos  x  will  be  diminished 
by  2,  or  be  less  by  2  than  in  the  proposed  integrals. 

EXAMPLES. 

1.  Tofind/e^sin'.rf.,and/.«eos'.^.;  .  =  the  hyper- 
bolic  base. 

Since  A  =  ^,  we  have  log  A  =  log  e  =  l]  and  as  a  =  1  and 
7?i  =  2,  w^e  have 

/_   .        ,  e^  sin  a?  ,  .  .  .         2    ^    , 

e''  sm  xdx  =  -^«-^ —  (sm  «  —  2  cos  a?)  +  ■=  e""  +  const, 
O  '5 


400  INTEGRALS  OF  VAEIOUS   F0RM8. 

•i  r    -r  0  7  ^*  COS  X     ,  r.       •  V  2        , 

and    /  e^cos^xdx  =  — ^ —  (cos  87  +  2  sm  a?j  +  tt  ^*  -f  const. 

2.  To  find  /  S""  sin-  a?(^,  and  /  3^  cos^  xdx. 

Here  A  =  3  and  log  A  =  log  3  =  1.09861,  «&c.,  a  =  1  and 
m  =  2;  and  thence,  from  {a)  and  (J), 

/  3^  sin^  xdx  = 

3^  sin  a?       ,-      ^    .  ^  ,  2  3* 

(log  3)-  +  2^  (^"g  3  sm  a.  -  2  cos  x)  +  ^^^^  3. ,  ^  ^,  x  ^--3, 

and  /  3""  cos- xdx  = 

3^cosa;       ,,      „  n    •      N  2  3^ 

(log  Sy  +  2-  ^^"g  3  cos«.  +  2  sm  .)  +  ^--3y-_p^.  x  ^^3. 

3.  To  find    /  e""  sin  aa?(fa?,  and   /  e'^  cos  «;» Ja?. 

From         /  e^  sin  aa?^a;  =  -    /  g""  sin  «a?^  (aaj), 

and  /  6^  cos  <ia?c7aj  =  -   I  e^  cos  aa?«?  («a?) ; 

by  patting  ax  =z  y  ov  x  =  -  ^  \nq  have 

/  e""  sin  axdx  =  —^    /  e"  sin  y<fy, 

and  /  e^cos  axdx  =  —„   I  e^  cos  ydy. 

Hence,  from  (a)  and  (h),  by  putting  -  for  a,  we  have 

a> 


y_ 
r^.        dy             e"        /sin  v  \  1 

/  e°  sm  V  -4  =  -rrr^ I ^  —  cos  ?/     - 


+  const, 


ILLUSTRATED   BY   EXAMPLES.  401 

r  ^-  dy  e"        /cos  y         .      \  1 

and       /  e""  cos  y  ~  z=z  — — I ^  -f  sm  ?/ 1  -  +  const. 

J  ^  a-        /ly      .2  ^    «  '^1  a 

By  re-substituting  the  value  of  ?/,  we  shall  have 

e-^  sin  ascdx  = 5  (sin  ax  —  a  cos  ax)  +  const., 

and  /  6  ^  cos  aa?c?aj  = 5  (cos  ax  -h  a  sin  aa?)  +  const. 

J  1  +  a^ 

4.   To  find      /  e^  sin'^  xdx,     and      /  e^  cos^  xdx. 

From  the  tables  given  at  pp.  77  and  78,  we  have 

.0  8  sin  a?  —  sin  3a?  ,  „  3  cos  a?  +  cos  3a? 

sm.-^  X  = ,     and     cos^  x  = ^ , 

4  '  4  ' 

which  reduce  the  integrals  to 

/  e^  sin'^  xdx  =  -    /  6^  sin  xdx  —  j   I  ^^  sin  Sxdx^ 

and   /  e^  cos''  xdx  z=z  -    I  e^  cos  xdx  +  t   I  ^^  cos  3a'cZa?.    • 

By  taking  the  integrals  by  (a)  and  {h),  agreeably  to  what 
has  just  been  shown,  we  have 


/  e^  si 


sin'^  xdx  = 
-Q-  (sin  x  —  cos  ^)  ~  -jK  (sin  8a?  —  3  cos  8a;)  +  const, 


/. 


and  /  e^  cos^  xdx  — 

8e^  .  e^ 

-^  (cos  X  +  sin  a?)  +  -—  (cos  8aj  +  3  sin  Sa?)  +  const. 

(10.)  We  will  now  show  how  to  find  the  integrals  of  dif- 
ferentials into  whose  differential  coefficients  enter  arcs  with 


402  INTEGRALS  OF  VARIOUS  FORMS. 

algebraic  functions  of  the  arcs.     Thus,  to  integrate 
/  X  (sin~^  xY  djc      and      /  X  (cos~^  ic)"  o?.r, 
X  being  an  algebraic  function  of  the  arc,  it  is  clear  that  we 
may  put   /  X.dx  =  Xi ,  and  thence,  integrating  by  parts,  get 

fx (sin-^  xf  dx  =  (sin-^  xf  X,-  nf{sm-'  xy-'  X,     /^    ..  , 

, . ,  ,         .      r   x,dx 

which,  by  putting  y  -—-—^    gives 

{sm-'xy-'X,-{?i-l)f{sm-'xV-'X,^^,; 
and  so  on  to  any  required  extent  in  this,  and  such  forms  as 
JX  (cos -^  a-)" dx,  Jx  (tan-^  xf  dx,  JX  (cot"'  xf  dx . . .  (A). 

EXAMPLES. 

1.    To  find    I  X  sin~'^  xdx     and     I  x  cos~^xdx. 

/x- 
xdx  =  -,     which  gives 

X" 

Xi  =  —  ;   consequently, 

dx  ,  1  /*     ij^dx 


f^'^Ai^)    ^""^^^"'    If 


|/(l-ar^)  2  J   1/(1 -a^')' 

and  thence 


fx  sin  -  >  Xo,  =  «_i!il.^  _  ip  (1  _  ,^)  -  ij, 

4/(1 -fJ) 


ar'  sin-^'»       1        ,,         ^,        1  /*       dx 


2aj2 1  1 

— - —  sin~^  ^  +  J  a?4/(l  —  ar')  +  const. ; 


ILLUSTRATED   BY   EXAMPLES.  403 

and  in  like  manner, 

/2a!^  +  1  1 

X  0,0^-^ xdx  —  V —  cos ~^ a?  —  j  x\/{l  —  x'^)  -h  const. 

2.  To  find   /  a?"^  -  ^  sin  -  ^  xdx     and    /  x "'  - ^  cos  -  ^  xdx. 

x"^~'^dx  =  — ,  we  shall,  by  integrating  by  parts, 

from  (A)  get 

r        ,    .       ,     y         sin  - '  a:'!^?"'        1    T  «        dx 

I  x"'  ~'-  sm ""  ^  xdx  = /  a?"'  -—, — -— ^ , 

J  m  mJ        |/(  —  X-) 


■X      r  .     A  17         co^-^x^x"^        1    r  „.        dx 

and     /  a?"'-^  cos~^  xdxz=^ -\ \  x^  — ^ 

J  m,  tnJ         4/  (;1  —  X' 


we  also  Lav 


dx  \/{l  —  ar)x'^-'^      m  —  lr        o       dx 


r..m       dx        _  4/(1  -  x^) X--'  m-l  r 

J        4/(1 -^'')~             'i^'  m    J^        |/(1-^'')' 
and  by  tbe  same  process 

dx           _l^{\—Xr)X'^~^  ^^^'-^    /*    ,n-4          ^^ 


C^..-.     dx      ^  v{i-xr)x-~^  _  m-s  r      ,__ 


and  so  on  to  any  required  extent.  It  is  hence  clear,  that  if 
m  is  an  odd  positive  integer,  the  complete  integrals  of  the 
proposed  integrals  will  be  algebraic ;  while  if  r/i  is  an  even 
positive  integer,  they  will  be  reducible  to  circular  arcs  or  be 
dependent  on  them. 

Remark. — It  is  easy  to  perceive  that  like  conclusions  are 
applicable  to  the  integrals 

-'^  ian-'^  xdx     and      I  x''-^  tan.-^  xdx. 


I  x"'-'^  tan-'^xdx     and      /  ; 


404  INTEGRALS  OF  VARIOUS  FORMS. 

3.  To  find 
/  — ;;;^in—  =   /  sin  -  ^  ica;-"^^  dx    and      /  cos"^  xar"'-'^  dx  , 

/x~^ 
x-""-^  dx  = =  Xi,     these  integrals 

become 

/,         ^    .  ,              sin-^«,aj-"*       1   /•          dx 
sm-'xx-"'-^dx= i +  -  /  -i;r-^7T ^^ 
m               mJ  x"'  4/  (1  —  ar)  ' 


,         /*cos ~ ^  xdx  _       cos ~^x       1    r 
J      a."*+"^'~  ~         irix""         mJ  x' 


dx 


If  m  is  a  positive  integer  not  less  than  2,  we  shall  have 

m  —  1  m—  w  ^  ^ 

bj  putting  for  ?i  its  value  2.  By  changing  —  m  into  —  771+2 
we  shall,  in  the  same  way,  have 

m  —  3  7)%—ZJ  ^  ' 

and  so  on.     Hence,  if  m  is  an  odd  positive  integer,  we  shall 
C       dx  T      1  +  i'(l  —  •'»')      n 

which  will  enable  us  to  find  the  integrals  corresponding  to 
any  other  odd  positive  integer,  while  it  is  manifest  from 
what  is  done  above,  that  when  m  is  an  even  positive  integer, 
the  integral  is  algebraic,  and  can  be  exactly  found  by  the 
preceding  process.     Thus 


ILLUSTRATED   BY   EXAMPLES.  405 

-  2  ^  .„  -  2 


sm-^x       1    (1  -«-)^ 


,2  c,  „„        +  const; 


^x"  2  X 

and 

r  1  w  COS-^'2?         1      (1  —  9?^)"^    ,  , 

/  QO^-^x^x-^dx=  —  — ^-.^ \-  ^ +  const. 

Remark. — It  is  clear  that  we  may,  in  mucli  the  same  way, 

«    ,  f*  isin~^xdx         ,       rGoi~^xdx 

fl°d  y  — ™-T^     and     j-~„-iT-. 

4.  To  find    I  {sm-'^x)'dx  and    I  x-{co8-'^xydx. 

Since  X  =  1  we  have    /  Xc/^  =  a?,  and  thence,  from  (A), 
I  (sin-^  xf  dx  =  (sin-^  «)^  a?  —  2    /  sin-^  a?,  '  - — ^  , 

and  in  like  manner  from  X  =  a?-,  we  have   /  x^  dx  =  ~  ^  and 
thence  from  (A), 

J  ar  (cos-^  xfdx  = —^ h  o  /  cos"'  a?— yr- 

We  also  have       /  sin-^  ^ 

^  V  (1    —   i^  ) 


xdx 


„ ....,,,         ..,         dx 

sm 


dn-i  a?  |/(1  -x")  +  f  ^{\-a?) 


~  —  sin-^  a?  y  (1  —  a?^)  +  cc  ; 
consequently,  we  shall  have 

J  (sin-i  xj  dx  —  (sin-^  aj)"a?  +  2  sin-^  x  |/(1  —  ar^- 2a; + const 


406  INTEGRALS   OF   VARIOUS  FORMS. 

From  IV.,  at  page  382,  we  have 

""                3  3  * 

and  thence  /  cos"^  a?.2r' — t^^ :r, 

=  -  cos-^  X  ^^^^—-—^-  +  -  |/(1  _  x")!  - 
consequently, 

<>  £P*  2.'?/ 

g  cos-^  x[^{l-x')x'-2  v'(l  -  ^')  ]  -  36  +  36  +  ^^^s*- 


5.  To  find    /  X  tan"^  xdx  and    /  a?  cot~^  xdx, 

(Tfic  =  -  ,  we  have 

r    .        ,   .,  ,         (tan-»aj)»aj=         A     _i      ^'^^^ 
/  a;  (tan-'a?)^  dx  =^ ^-- /  tan^a; -^ 

and 

r    ,        ,   ,,  ^         (cot-' ajVa;^  r    ,   ,      irWa? 

If  at  IV.,  at  p.  382,  we  put  a  =  5  =  1,  n  =  2,  and  7n  —  2, 
it  will  give 


ILLUSTRATED   BY 

EXAMPLES. 

407 

r  xHx 

=  fx'  (1  4-  xY 

'^  dx  ^=^  X  — 

r  dx 

or 

taking  the  differentials, 

x^dx           , 

r— 2     —    dx    — 

1    +    «^ 

dx 

which  can  be  found  more  simply  bj  actual  division. 
Hence,  by  substitution,  we  have 

/  X  (tan~^  a?)^  dx  = 


(tan-^a^y^^          P        i     /      ,      A       i         ^^'^ 
•^^ —-^ /  tan-^  xdx  +    /  tan"^  x  z ^ , 


and 


/  a7(cot-  ^  x)-dx  = — —  +  /  cot-  ^  xdx  —  /  cot" '  x  rr— — 2 ; 

and  since 

/  i^n-^xdx  =  tan-^  Xi^x  —  -  log  (1  +  x"^), 


dx 


and  r~7~^  ~  ^^  (t^^~^  x)  =  —  d  (cot-^  x). 


we  shall  have  finally 

I  a?(tan-^  xydx  = -— ^ h  ^^ — ^ 

tan-^  XiX  +  -  log  (1  +  ar^)  +  tan"^  a?  -f-  2  +  const, 
and 

r    /    .  1    NO  7         (cot-*a?Viz;'^       (cot-^  ojV 
y  a;  (cot-i  xy-diB  =  ^ ^-^—  -f  ^^ — ^—^  + 

cot-^  XiX  +  -  log  (1  +  x")  +  const. 

(See  Lacroix  "  Calcul  Integral,"  pp.  95  and  06.) 


408  INTEGRALS   OF   VARIOUS   FORMS. 


Eemarks. — It  is  manifest,  from  what  has  been  done,  that 
to  find  an  integral  of  the  form 

\a'  +  h'  cos  z)  dz 

{a  -{-h  cos  zY    ' 

we  ought  to  represent  it  by  the  form 

A  sin  z  r{B  +  C  cos  i)  dz 


r- 


J    (a 


{a  +  b  cos  zy-'^       J    {a  +  b  cos  2)"-^ 

For  by  taking  the  differentials  of  these  equals  we  have 

{a'  +  1/  cos  z)  dz  _ 
{a  -\-b  cos  2)"      ~~ 

A  cos  zdz  ('^~  ^)  A  J  sin^  zdz         (B  +  C  cos  2)  dz  ^ 

(a  +  b  cos  s)"-^  {a  ^-  b  cos  z^  (oTTcosT)"^  ' 

or,  by  omitting  the  common  factor  dz  and  a  simple  reduc- 
tion, we  have 

a'  +  Z>'  cos  s  =  A  cos  2  (a  +  ^  cos  s)  +  {n  —  1)  A5  (1  —  cos''  2) 
+  (B  +  C  cos  z){(i-^b  cos  2), 

or    a'  —  (;i  —  1)  A5  —  Ba  +  {b'  —  Aa  +  Bb  —  Ca)  cos  s 
-  [Ab  +(n-l)Ab+  Cb]  cos'z  =  0, 

which  must  clearly  be  an  identical  equation,  and  be  satisfied 
so  as  to  leave  cos  z  and  cos^  z  arbitrary ;  consequently,  we 
must  have 

a'  -(n-  1)  Ab-Ba  =  0,     b'  -Aa-Bh-  Ca  =  0, 
M-{n-l)Ab-h  Cb=0. 
From    the    last    of   these   equations   we   immediately  get 
C  =  (;i  —  2)  A,  which  reduces  the  second 

J'_ Aa-B5-  {fi-2)  Aa  =  0,  or  b'-~Bb-  (n-l)  A=0, 
which  gives  B  =  j ; 


r 


ILLUSTRATED   BY   EXAMPLES.  409 

and  thence  from  the  first  equation  we  have 
.   _        ah'  —  ha' 

{n-l){a}-hy 
and  of  course 

B=«4^  and  c=(;-?,<;^'-y.. 

ar  —  0^  {n  —  I)  (<2-  —  h^) 

Hence,  from  the  substitution  of  these  values  of  A,  B,  0, 
in  the  assumed  integral  equation,  we  shall  have 

*{a'  +  h'  cos  z)  dz  _  {ah'  —  ha')  sin  2 

{a-\-h  cos  zy    ^  (fi  —  1)  («■'  —  b')  (a  -{- h  cos  zf  "^  ^ 

1  ri{n-l){aa'-hh')-^{n-2){ah' -ha')  cos  z] 

{n-l){a'-¥)J  ^  {a +  h  cos zY~'  ^ 

so  that  the  complete  integral  is  reduced  to  that  of  another  in 
which  n  is  represented  hj  n  —  1;  consequently,  if  7i  is  a 
positive  integer  greater  than  1,  we  shall,  by  successive  rep- 
etitions of  the  process,  finally  reduce  the  integral  to  that  of 
an  integral  in  which  n  is  equal  to  unity,  or  to  the  form 

\p  +  5'  cos  z)  dz 
a  -\-h  cos  z 

(see  Lacroix,  p.  109) ;  noticing  that  this  integral  is  reducible 
by  division  to  the  more  simple  form 


/ 


q^    ,    hp  —  aq    (*         dz 

OS 


J  "^  ^   h      J  ^TFT 


cos  z 


If  (with  Lacroix,  at  p.  106)  we  put  cos  z  = ^ ,  we  shall 

X   ~\~  OS 


get  f       f       '=2f- 

°  J  a  -\-  0  cos  z  J  a 


dx 


+  J  +  {a-h)x'' 
whose  right  member  is  clearly  of  an  integrable  form.     Since 


cos  z  —  cos^  -  —  sin^  ^ , 


we  shall  have 

18 


410  INTEGRALS  OF  VARIOUS  FORMS. 

/dz  _       r dz 

a  +  J  COS  2  "~    /       ,   ,  /      7^         .  .z\ 
%/    a  +  0  I  cos'  -  —  sin'^  -1 


_    r dz 

J  cos^  I  Ua  -\.h)  +  {a-h)  tan'  |) 

Va  —  b  di 
2 f_±±]Ll 


/a  —  h  dz  .2 

— —  o-  -^  cos'  - 


a  +  5  2 

in  wliich  a  and  h  are  supposed  to  be  positive,  a  being  greater 
than  h.     Since 

o  +        1  a/^  "~  ^  4.      ^        ^        1  sin  s  4/  («'  —  J") 

2  tan-^  y y  tan  -  =  tan"^  — ^    ^  ^ ^ , 

a  +  J         2  5  +  acos2' 

we  liave 

r         dz  1  ^        ,  sins  4/ (a'  — J') 

/  — r-T =  -77-1 F\  *^"~       r    . ^  +const.: 

J  a  +  0  cos  s        4/  (a^  —  6-)  6  +  «  cos  s  ' 

noticing,  that  the  same  integral  may  also  be  expressed  by 
either  of  the  forms 


/ 


dz  1  .        sins  4/ (a' -Z>') 

,  1 =  -m> 7^  sm  -    f-^ ^  +  const., 

a  -\-  0  cos  z         \/  {a'  —  0')  a  -\-  0  cos  z 


dz  _         1  _  J  J  +  a  cos  z 


7  =  — r-s fT,  cos  ~  ^ y h  const. 

a  +  b  cosz        \/or  —0^  a  +  0  cos  z 

(11.)  We  have  shown,  at  p.  262,  that  every  differential 
expression  containing  a  single  variable,  admits  of  an  integral 
of  either  a  diverging  or  converging  form,  by  integrating  by 
parts,  as  in  John  JSei'tiouillis  Theorein.  We  have  also 
applied,  at  the  same  page,  the  Theorem  of  Maclaurin  to  ob- 
tain series  of  more  rapid  converge ncy  than  can  often  be  done 


ILLUSTRATED   BY    EXAMPLES.  ^^  4:11 

by  tlie  aid  of  the  Theorem  of  Bernouilli ;  and  from  the 
problem  at  p.  266  we  have  obtained  formulas  for  the  compu- 
tation of  such  integrals  by  series  of  any  degree  of  con- 
vcrgency  that  may  be  required. 

Because,  in  what  has  been  done,  the  series  have  been 
supposed  to  be  arranged  according  to  the  ascending  powers 
of  the  independent  variable,  we  now  propose  to  show  how 
to  apply  series  to  find  integrals  when  the  series  are  arranged 
either  according  to  the  ascending  or  descending  powers  of 
the  independent  variable. 

1st.  To  find  the  integral  of  a  proposed  differential  by  a  se- 
ries, it  is  manifestly  necessary  to  convert  the  differential  co- 
efficient of  the  differential  of  the  independent  variable, 
according  to  the  known  methods,  into  a  series  arranged  either 
according  to  the  ascending  or  descending  powers  of  the  in- 
dependent variable  ;  then,  to  multiply  the  terms  of  the  series 
by  the  differential  of  the  (independent)  variable,  and  to  add 
an  arbitrary  constant  to  the  sum  of  the  integrals  of  the  pro- 
ducts, for  the  integral  of  the  proposed  differential. 

It  is  manifest  that  the  sum  or  generating  function  of  the 
series  thus  found  will  be  the  finite  integral  of  the  proposed 
differential 

EXAMPLES. 

/dx 
A~~ — ^  ^y  ^  series,   arranged 

either  according  to  the  ascending  or  descending  powers  of  x. 

By  dividing  dx  by  1  -|-  a?^,  when  1  is  taken  for  the  first 
term  of  the  divisor,  we  have 

/  :j — '- — 2  =    /  {dx  —  x'dx  +  xdx^  —  xdx^  +  &c. 


+ 

a^       ^       X 

"3"^  5"~  7 


-^  +  -F —  TT  +  &c.  4-  const, 


412  INTEGRALS   OF    VARIOUS   FORMS. 

for  the  development  when  the  series  is  arranged  according 
to  the  ascending  powers  of  the  variable.  And  by  taking  ^ 
for  the  first  term  of  the  divisor,  we  have 

/dx      _    Cidx       dx      dx      dx  \ 

for  the  form  of  the  integral,  when  it  is  arranged  according  to 
the  descending  powers  of  x.  To  find  the  constant,  we  re- 
mark, that  X  being  the  tangent  of  an  arc  whose  radius  =  1, 
it  is  clear  that,  supposing  the  arc  and  tangent  to  begin  to- 
gether, the  constant  in  the  first  integral  must  equal  naught, 
and  the  integral  becomes 

dx  ^      x'      x^ 

while,  by  supposing  x  to  be  unlimitedly  great  in  the  second 
integral,  it  will  clearly  be  reduced  to  the  constant  in  its  right 
member,  since  the  terms  which  involve  x  must  clearly  be 
rejected  on  account  of  the  infinite  value  of  a?,  and  at  the 

same  time  /  -^ r  must  equal  ^ ,  the  length  of  the  arc  of 

the  quadrant  of  a  circle  whose  radius  ==  1 ;  consequently, 

the  constant  in  the  second  integral  equals  '- ,  and  the  integral 

becomes 


/i 


/ 


dx  TT  1  1  11  - 

a.-2  +  1  ~  2       x^  Zx^       bx'^  Ix'      ' 


/dx 
— -,■:, s^   =  si 
\/{l-x-) 


sm-^ic. 


in  a  series  arranged  according  to  the  ascending  powers  of  a?, 

1      ,         1.3  , 

=  ^+2:3^  +  2A5+'^^-' 


ILLUSTRATED   BY  EXAMPLES.  413 

wHcli  needs  no  correction,  supposing  the  arc  and  sine  to 
commence  together. 

Since  the  binomial  theorem  gives 

^  .-.        o.-i      ,      a?-'      1   3    ,      1   3   5    6 

(1-^^)   ^  =  1+      +x' +-.-.- x'+,kc., 


y{l-x')      ^^     ^'  '    2  '   2'4       '246 

we  shall  have 

1      ,         1.3       ,        1.3.5      , 

sm-^o.  =  ^  +  O  ^  +  27475  ^  +  2747  677  ^  +'  ^''' 

which  needs  no  correction,  supposing  the  arc  and  sine  to 
begin  together. 


/dx 
—7rrT-^\  "=■  ^^g  ['^  +  1/(1  +  «^')]  +  CJ' 

in  a  series. 


Because 


1  .       1.1.3,      1.3.5   ,   ,     , 


4/(H-a5'^)  2       '   2.4  2.4.6 

we  shall  have 
log  [x+V(\  +  '.<?)-\=x-^^x>  +  i-^^  a^-  i||-  + ,  &c, 

which  needs  no  correction,  supposing  the  integral  to  com- 
mence with  X ;  which  we  clearly  may  do,  since  ic  =  0  gives 
log  1  ==  0,  as  it  ought  to  do. 


/dx 
Ti7~i~Zrr\  ~  ^^^  '-^  "^  \/{x^  —  1)]  + 
a  series. 
Since  \/ {x^—  1)  =  X  y  ll A^  we  clearly  have 

4/  (»2  -  1)       x\        xV 

1    _L     1-^       1.3.5 

aj'^2c^"^2.47?+  274.6^^"^'        ' 


414  INTEGRALS   OF   VARIOUS   FORMS. 

consequently,  we  shall  have 

and  by  putting  a?  =  1  in  this,  since  log  1  =  0,  we  have 

1  J.^3 hJjL^   -    Xr 

^-~2.2      2.4.4       2.4.6.6      '*''•' 


and  thence 

i/^^rz.i\— ^  _  , 

2.2  ^  2.4.4  '   2.4.6.6 


log  {x+V^u^-  1)=  A  +  o^i~T  +  o^^#A  +  &«• 


1  1.3  1.3.5 

+  ^og  aj  -  2^-,  -  ^^^-4  -  274:6.  6^«  +'  ^^- 

=  loD^  (a  +  a?)  into  a  series. 

a  +  a?  °  ^  ^ 

dx  /I        37         a?         ar  \ 

Since =  ( b  H — <, ^  +  &c. )  dx,  we   shall 

a  +  a?        \a       a-       a^       a*  / 

lyt  /y»2  rti°  /yi^ 

have    log  (a  +  x)^'--~  +  ^,-^,  +  &c.  +  C, 

by  putting  a?  ==  0  in  this,  we  have  log  a  =  C;  consequently, 
we  have 

log  («  +  ^)  =  log  a  4-  -  -  2-.  +  3«-3  -  4-7.  +.  &=• 

Remarks. — It  is  easy  to  perceive  that  this  deYclopmont 
can  be  immediately  obtained  from  log  a,  by  changing  a  into 
a  +  a?,  and  then  developing  log  (a  +  a?)  according  to  the 
ascending  powers  of  ^,  by  Taylor's  theorem. 


6.  To  find  the  integTaiy|/'lz:_^Z:  .j^  ^  f^~=^  ^^^ 

J.  —  a?  V     f  L  —  X' 


m  a  series. 


ILLUSTRATED   BY  EXAMPLES.  415 

Bj  converting  |/(1  —  e-x^)  into  a  series  arranged  according 
to  tlie  ascending  powers  of  x,  we  have 


//v 


/dx 
——, r-  =  sin-^a?,  and  that   formula  IV.,   at 
|/(1  —  a?-)  ' 

p.  382,  gives 
r      xHx  r  2,.        2x-i^  a?4,/(l— a?-)      1   .     . 


and 


/x'      1     3    \    ,,,         ,-13.      , 

=  -  (4  +  2  •  i  ^/  ^(^  ~  ""')  "^  2  •  4  ^'''~  ^' 

and  fx'{l-a^)-^dx  = 

\6  +4-6^  +  2*4:-6^r  ^      "^  +2-4-6 
and  so  on ;  by  collecting  these  results,  we  shall  get 


5    .      , 


J  V  Y^'aT        "^  sin-^a?  +  ^\^2^^'^  ^^  ^~  2^^^~    j"^ 
e*    f/l    ,       1  3    \    ,,,         .,        1  3    .      ,    1        1.3«« 


416  INTEGRALS  OF   VARIOUS   FORMS. 

for  the  integral ;  whicli  needs  no  correction,  supposing  it  to 
commence  witti  a*. 

Remarks — It  is  easy  to  show  that  the  j  -receding  integral 
represents  an  arc  of  an  ellipse,  reckoned  from  the  extremity 
of  its  minor  axis. 

For  let  y  =  -  |/  (a*  —  x^)  represent  the  common  equation 

of  an  ellipse,  then  if  e  equals  the  ratio  of  the  distance  of  the 
focus  from  the  center  to  the  semi-greater  axis,  we  shall  have 
J  =  a  |/(1  —  ^)  for  the  half  minor  axis,  and  the  equation 
of  the  ellipse  reduces  to  y  =  |/(1  —  ^)  i/(«^  —  ar*) ;  'whose 

differential  gives      dy  =  —       .^^ — ^  ox  -  •     Hence 


dy'  +  d^  =  d^^=  (^=^4^^  +d^.  or  dz  =l/^5'<fe, 
^  a'  —  x^  a-  —  nr 

which  agrees  with  the  preceding  differential  equation  when 
a  =  1,  and  it  is  clearly  the  differential  of  an  arc  of  the 
ellipse  reckoned  from  the  extremity  of  the  minor  axis. 

If  we  put  a?  =  a  sin  0  we  shall  have  dx  :=^  a  cos  <f)d<l>^ 
wliich  reduce  the  differential  equation  to 


dz  ^=^  a  ^\  —  ^  sin^  <^  x  ci?</> ; 
if  we  put  a  =  1,  the  half  major  axis  =  1,.  and  we  have 


dz  ^  y  \  —  ^  sin^  <^.  c?0 


or  representmg  rl— Vsin^  0  ^J  ^j  we  shall  have 

which  is  an  elliptic  function  of  the  second  kind,  according 
to  the  notation  of  Legendre.  (See  p.  19  of  his  Exercises, 
"DeCalcul  Integral.") 


ILLUSTRATED   BY  EXAMPLES.  417 

Q.          .  <,  ,        1  —  COS  20  , 

Dince  sm-  9  = ,  we  have 

or  putting  20  =  6^,  we  have  c?0  =  —  ;  consequently,  we  sliall 

have 

dz=z  |/(1-  6=  sin^  0)  d<^  =  r  -y--  X  |/(l  +  ^372  cos  ^^t^^, 

or  putting ^  =  ^1  w®  ^^^^^  -^^^'^ 

/o  ^2 

6^2  =  y  — — -  1/1  +  c  cos  d  (Id. 

o 
Bj  tlie  binomial  theorem, 

4/  (1  +  c  cos  0)  =  1  +  ^  c  cos  0  —  ^  c^  cos^  ^  "^  Tft  ^^  ^^^^  ^  — 

consequently,   multiplying  the   terms   of  this   by  d6^    and 
taking  the  integrals  from  0  =  0  to  0  ==  tt,  we  shall  have 

/V(i+.oos«)d.=.(i-  y-^,0*-  Ssr'-  ^'=-)' 

which  gives  the  quadrantal  arc  of  the  ellipse,  reckoned  jfrom 
the  extremity  of  the  minor  axis, 

*^       2  2 

A_l_il__Jl^ ^5    _^L-_&e) 

\        16(2-«7      1024  (2  -  <?'')«      16384  (2 -«y  7" 


418  INTEGRALS  OF  VARIOUS  FORMS. 

If  we  apply  tliis  formula  to  find  the  perimeter  of  the 
ellipse,  whose  major  and  minor  axes  are  12  and  D,  we  shall 

have  .=  =  1  -  J  =  1  =  0.4375, 

and  thence  ,- 5  =  0.28 ; 

hence,  we  easily  find  0.99501,  for  the  sum  of  the  first  three 
terms,  within  the  parentheses,  of  the  preceding  series ;  and 

since  V  ^-^'  =  1/0778125  =  0.88388, 

it 

we  have  0.88388  x  0.99501  =  0.87947. 

This  is  the  same  result  that  Mr.  Young  has  obtained  at  p. 
116  of  his  "Integral  Calculus,"  from  the  formula  at  p.  415, 
when  taken  between  the  same  limits,  or  from  a;  =  0  to  a?  ==  1, 
which  gives 

'l^^       ^A  ^      1.1.1.3   ,  _  1.1.1.3.3.5   6  _  xr   \ 
2  V       2.2  ^  ~  2.2.4.4  ^        2.2.4.4.6.6  '        ^^7 

for  the  length  of  the  quadrantal  arc  of  an  ellipse  when  its 
half  major  axis  is  denoted  by  1  or  unity.  Mr.  Young  ob- 
tained his  result  by  calculating  the  first  eight  terms  of 
this  series,  whereas  the  first  three  terms  of  our  series 
have  given  the  same  result,  which  clearly  shov/s  thot  our 
formula  is  far  more  convergent  than  the  preceding  for- 
mula, which  is  the  formula  commonly  used  for  the  compu- 
tation of  the  elliptic  quadrant.  It  is  easy  to  perceive  that 
we  shall  have 

^  X  24  X  0.87947  ^  66.31032 

for  the  perimeter  of  the  ellipse,  which  difi^ers  but  little  from 
Mr.  Young's  result 


DEVELOPMENTS  IN  SERIES.  419 

(12.)  We  now  propose  to  show  liow  to  apply  series  to  the 
computation  of  integrals  of  the  forms 

JX!^dx'^,fx''dx'',     &c., 

which  have  been  partially  considered  at  pp.  312  to  315. 

/dx^                             1 
z 5 ,  we  convert  q ^  into  a  series, 
1  —  ar                        1  —  ar  ' 

r-^— ^  =  1  4-  a^  +  aj*  +  a?«  -f ,  &c., 
1  —  ar 

and  thence  get 

/        ^   .  =  /  dx  J  {dx  -\-  x^dx  -\-  x^dx  -{-  x^ dx  -\-  &c.) 

=  I  Ixdx  -{-  -  dx  -\-  ^  dx  +  &G.  -\-  Cdx) 
we  also  have 

=  J  ^y  ^K''  ^  2T3^  -^  274.5^  +  2.-077^  +  ^'-  +  ^) 
r,    (x^     1.1.1    ,     1.3.1.1    ,     1.3.5.1.1   „     ,        r.       ^A 

-2.3  "^2.3.4.5  "^2.4.576.7  + 2.4.6.7.8.9  "^'^''•'^  2   ^^-^^^  5 

and  it  is  evident  that  each  successive  intesrration  introduces 
a  new  arbitrary  constant ;  consequently,  the  number  of  arbi- 
trary constants  must  equal  the  number  of  successive  integra- 
tions of  the  proposed  differential. 


420  DEVELOPMENTS   IN   SERIES. 

In  like  manner,  if  we  have 

/  ico^  —    I  dx  I  dx  I  xdXj 
by  representing  it  by  y,  we  have 

y  =  fdx  fdx  ( J+  C)  =fdx  (I'  4-  C.^^  4-  C') 

=  ^  +  ^  +  c^^  +  c- 

which  clearly  represents  a  curve  of  the  parabolic  kind  of 

d^v 
the  fourth  order.     Hence,  if  we  h'ave  -j-~  =  X,  representing 

cix 

I  Xdx  by  Xi ,    /  ^idx  by  Xg ,  and  so  on,  we  shall  clearly 

dh^^  =  f^idx  +  fcdx  =  X,-\-  C,x  +  C2, 

and  so  on,  to  any  required  extent 

2.  It  is  manifest  that  these  processes  are  applicable,  whether 
the  expressions  to  be  integrated  are  of  algebraic  or  transcen- 
dental forms.     Thus  we  have 

/  cos  xds^  =    I  dx   I  dx    I  cos  xdx  = 

/C  .  Caj' 

dx  J  (sin  a?  +  C)  c?a?  =  —  sin  a?  +  -—-  +  Q>'x  +  C; 

also  J  e^dx^  =  fdx  fdx  {e"  +  C)  = 

fdx  (e«  +  Ca?  +  CO  =  ^"  +  -^  +  C  cc  +  Q'\ 


DEVELOPMENTS   IN  SERIES.  421 

3.  Mr.  Young,  at  p.  91  of  his  "Integral  Calculus,"  gives 

(/"- Xc&»-)  ^^^  +  . . . .  (/X.&)  j£lU  + 
^^r27...7i  "^  \dx)  1.2. ..."  +  1"^ 


\^ji:2 -^'  '^'*''-' 

in  whicli  tlie  development  is. made  according  to  the  ascending 
integral  powers  of  a?,  by  Maclaurin's  theorem;  (  /  X(ia?"j 
denoting  the  last  of  the  arbitrary  constants  according  to  the 
preceding  methods  of  development,  I  /  XcZa?"~M  denot- 
ing the  last  constant  but  one,  and  so  on  until  there  are  no 
arbitrary  constants;  noticing,  that  the  terms  within  the 
parentheses  stand  for  the  values  of  the  corresponding  quan- 
tities, when  a?,  in  them,  is  put  equal  to  naught 

EXAMPLES. 

—7(-T- K  according  to  the   ascending 

powers  of  x. 

Since  the  binomial  theorem  gives 

^        =l-^.^^  +  ^a^^-   &c., 


Vl+x-'  2  2.4 

the  development  is 


422  DEVELOPMENTS  IN  SERIES. 


2.3.4.5.6    '    2.8.4.5.6.7.8 


x^         ^   X 


as  given  by  Mr.  Young. 

2.  To  develop    /    sin  xdx^   according    to    the   ascending 
powers  of  a?. 

sin  xdoi^  =  C3  +  CiX  -{-  Ci  ^  -f  cos  a? 

=  cos  i»  +  Ci  2    +  Cgaj  +  C3, 

as  in  Young. 

d*v 

3.  To  develop  -r^  according  to  tlie  ascending  powers  of  x. 

Here,  we  have 

y=:C,+  C,x  +  C,"^  +  C,^', 

which,  in  the  language  of  curves,  denotes  a  parabola  of  the 
third  order. 

4.  To  develop    /  e^'d.x^. 


Here 


/3  Qfi 

e'di^  =  C3  -h  C.x  +  Ci  2  +  e\ 


(13.)  We  now  propose  to  show  the  use  of  arbitrary  con- 
stants in  finding  definite  integrals  by  series  or  otherwise. 
According  to  what  is  shown  at  p.  265,  the  notation 

dx  n 


/; 


i/il-x")       2 


DEVELOPMENTS   IN   SERIES.  423 

signifies  tliat  the  integral  being  taken  from  x  =  0  to  £c  =  1, 

gives  ^  =  one-fourtli  of  the  circumference  of  a  circle  whose 

radius  —  i,  for  the  result  or  value  of  the  integral  contained 
between  the  preceding  limits  ;  and  a  like  notation  is  to  be 
used  in  all  analogous  cases  of  definite  integrals. 

If  we  take  the  integrals  indicated  in  example  7,  at  p.  386, 
from  X  =  0  to  x  =z  1^  when  m  stands  for  an  odd  or  even 
positive  integer ;  then,  for  7n  odd,  we  have  the  results 

r^      xdx       _        r^     xHx       _  2      r^     xMx       _  2.4 

J  o|/(l-a.'^)  "~    '  y  o|/(L-^  ~V  J  0,^/(1 -a;-)  ""  875 ' 

r^    xHx      _  2.4.6  r^x-'^'^Hx  _      2.4.6...  %i 

J  ^'^~x^)~ZJ7l J  o|/a-aj-)"~  3.5.7...  (2m  +  1)' 

by  using  2n  +  1  for  m,  and  by  proceeding  in  like  manner 
for  m  even,  we  shall  have 

r^        dx         _  n      r^     x'dx        _   1     ^ 

J  o7(r-^"^  ~  2'  c/  o^{l-x-)  ""2*2' 

/»!     x'^dx     ^  ij     TT      r^ Mx  1.3.5     n 

J  0  VT^^'  ~  274:'  2'  J  oTTa 


/ 


|/(l-aj'-)       2.4.6*2' 
^     x'"dx       _  1.8.5.7  ...  (2/1  —  1)     TT 


o|/(l-a;2)  2.4.6.8  ...  2;j.       '2' 

by  using  2n  for  wi. 

It  is  easy  to  perceive,  from  a  comparison  of  the  preceding 
values,  that  if  ?i  is  large  we  shall  have 

J  oTTi—^^  =  J  oTXT^^)  '''^'^^' 

1.3.5.7.  .  .  .  {2n  -  1)     TT  _       2.4.6 2?i 

''''        2.4.6.8  .  .  .72/1       •  2  -  3A7  . . .  .7(2rrhT)  ^^^^^' 

,    ,,'      n  2.2.4.4.6.6 2^.2/?. 

or  we  shall  have  ^  -  i.3.3.5.5.7...(2.  -  1).(2.+T)  ""'"'^^'^ 


424  DEVELOPMENTS  IN  SERIES. 

and  by  supposing  n  to  be  uulimitedly  great,  or  infinitely 
great,  we  must  evidently  have 

r  _  2.2.4.4 2;i.2??.  , 

2  -  1.3.3Ao....(2;i-iy(2;rrT)  ^^^""^^^^ 

for  the  length  of  the  quadrantal  arc  of  a  circle  whose  radius 
=  1  :  where  it  may  be  noticed  that  this  expression  seems  to 
have  been  first  discovered  by  Dr.  Wallis.  (See  Young,  pp. 
97  and  98,  and  Lacroix,  vol.  iii.,  p.  415.) 

Kemarks. — Mr.  Young,  although  he  has  with  reason 
objected  to  the  manner  in  which  the  formula  of  Wallis  is 
frequently  written  by  English  authors,  yet,  at  p.  97  of  his 
work,  he  has  written 

2.2.4.4.6.6.8.8      .       ""      -    ^    a  ^    2.2.4.4.6.6.8.8 
J — 7z~i-r~^ — f'  I-,  i-r  r^  t^     lor    ^ ,     instead  Cn.    z — „  „  ^ — ^  —  —  ,. , 
1.3.3.5.5.7.7.9.9  2'  1.3.3.0.6.7.7.9' 

which  is  its  proper  form  when  the  numerator  and  denominator 
each  consists  of  eight  factors ;  noticing,  that  the  numerator 

and  denominator  of  the  fractional  forms  of  -   must    each 

consist  of  the  same  number  of  factors  as  the  preceding  forma 

If  we  write  the  successive  approximate  values  of  - ,  after 

the  factors  common  to  their  numerators  and  denominators 
are  rejected,  we  shall  have 

2^  _  4  _         2^A4  _  64  _        19   2.2.4.4.2.2  _         81_ 
—  _  -  _  l.d,  ^  g  g^  -  45  ""    "^  45'  i.l.  1.5.5.7  ~    "^  175  ' 

2.2.4.4.2.2.8.8  _  -,  .  n   ,      2.2.4.4.2.2.8.8.2.2   _  -,  ^^   , 

1.1.1.5.5.7.7.9  ~  "^'  1.1.1.1.1.7.7.9.9.11  ~    *       '^' 

and  so  on.  From  these  results  it  is  clear  that  the  successive 
terms  approximate  very  slowly  to  the  IcDgth  of  the  quad- 
rantal arc,  the  last  result  being  correct  only  to  one  place  of 
decimala 


DEVELOPMENTS   IN  SERIES.  425 

For  anotlier  example,  we  will  show  how  to  find  the  de- 

—rr^ IT* 

Because  l  —  x^  =  (1  —  a^)  (1  +  x^\  we  shall  clearly  have 

r^ dx         _    f^       dx  ^     1  • 

f  r'       dx         /,       x"       ISx'       1.3.5    ,       ,    \ 

""  ./  0  V  (1  -  a?^)       2  y  0  ^(1  -  a?"')  "^  2.3  */  oy(l  -  a;-) 

1.3.5  /*      x'dx 

2.4.6  y   i/(l-a^)  '^' 

Because,  from  what  is  shown  in  the  preceding  examples 
we  have 


p       dx        _  TT     p 

/: 


Oy^dx  1        TT 


o|/(l-ar^)       2'^  o|/(l-a;'^-)  ~  2*2' 
x'^dx  1     3    TT 


|/(l-a^^)       2' 4*  2' 
and  so  on,  we  get  bj  substitution 

r       ^^  fi        /IV      /1-3V      /1.3.5\2      „     ]  T 

—  . 

oVx{i-x^) 


By  putting    y  —  2^x     we  get     dy  =  —,   ay^  =  U\ ; 
consequently, 


426  DEVELOPMENTS  IK  SERIES. 


A^-m 


V(l-3') 


by  putting  |  =  ^. 
Hence,  from  the  preceding  example,  we  shall  have 

which  is  twice  the  integral  found  in  the  preceding  example. 

(14.)  We  will  terminate  this  section  by  showing  how  to 
sum  series,  or  to  find  their  generating  functions,  by  means  of 
thie  preceding  principles. 

The  processes  here  proposed  seem  to  depend,  for  the  most 
part,  on  transforming,  by  means  of  the  integral  or  differen- 
tial calculus,  the  proposed  series  into  a  new  series,  or  in  find- 
ing a  new  series,  such  that  its  sum  or  generating  function 
can  be  found,  so  that  the  proposed  series  may  be  supposed 
to  have  been  derived  from  it. 

EXAMPLES. 

1.  To  find  the  sum  of  the  series 

^  =  a?  +  2^  +  8aj^  4- . . . .  ^-nx\ 
Multiplying  the  members  of  this   equation  by  —  and 
taking  the  integrals  of  the  products,  we  have 

I  8^-  ^=  I  {dx  +  ^xdx  -\-  Scc^dx  +  &;c.) 

=  X  +  a^  -{-  ar^  + 4- a?**, 


DEVELOPMENTS   IN   SERIES.  427 

which  is  clearly  a  geometrical  progression ;  whose  sum,  by 
the  common  rule,  clearly  equals  "-— ,  and  thence  we 

/dx       X ic"  "^  ^ 
s  —  1=  ^ — .     To  find  s,  the  sum  of  the  pro- 
X                1  —X 

posed  series,  from  this,  we  must  remove  the  sign  of  integra- 
tion   /  ,  by  taking  the  differentials  of  its  members,  which 
dx       dx  —  (ri  +  1)  a?"  dx  +  nx''  +  ^  dx. 

or,  by  a  simple  reduction,  we  have 

_x  —  (/?,  +  l)r6'"  + 1  +  nx""  +  2 


{1-xf 

2.  To  find  the  generating  function  of  the  same  series  con- 
tinued indefinitely. 

It  is  manifest  from  development,  that  we  shall  here  have 

s  =  -pr-- — ^2 ;  which  is  clearly  the  same  as  to  suppose  the 

definite  parts,  or  those  that  depend  on  7?,  in  the  preceding 

X 

sum,  to  destroy  each  other,  and  to  put  s  = 


3.  To  find  the  generating  function  of  the  series 


JJ.2  ^3  ^4 


^  + '^  +  2  +  2:3  +  2:3:4  + -^■■- 

Denote  the  sum  by  y,  and  we  shall  have 

y  =  1  +  X  +  ~   +  Ig+j&c; 

whose  differential  coefficients  give 
di/  x^        x^ 

;s  =  ^  +  *+2 +2r3  +  *''-  =  ^' 


428  DEVELOPMENTS  IN  SERIES. 

consequently,  we  shall  thence  get  -—  =  dx.     By  taking  the 

integrals  of  the  members  of  this  equation,  we  have  log  y  —  x^ 
which  needs  no  correction,  supposing  it  to  commence  with  a?, 
since  a;  =  0  gives  3/  =  1,  whose  log  =  0 ;  consequently,  put- 
ting e  for  the  hyperbolic  base,  we  shall,  by  the  nature  of 
logarithms,  have  y  =  e"^  and  thence 

6'  =  l+a,+  |  +_+,&c., 
as  required. 

4.  To  find  the  sum  of 

/;pn  +11  aj"  +  ^  aj"  ■•■  ^ 

*  ^  ^TTT  "^  2J{n~^~^)  "^  2.3.5 (71 +T)  "*"'  ^^ 

From  (J''),  at  p.  51,  we  have 

^=1  +  *  +  2   +2:3 +  2X4.5 +'*"■' 
and  putting  —  a?  for  a?  in  this,  we  also  have 

e~^  =  1  —  X  ■ 
whose  half  diiference  gives 

2  -  T  =  "' +  O  +  rsTlis +"^°-' 

and  multiplying  the  members  of  this  by  a?"-^  dx^  we  have 


„n  +  l 


'     O  Q/zK.     I     Q\    "T"   O  o  ./I  r:  /.,     ,     k\   +»  *^^-> 


n  +  1    '    2.3(w  +  3)       2.3.4.5(^1  +  5) 

since  the  method  of  integration,  explained  at  pp.    391   to 
S93,  reduces  the  equation  to 


,9  =:  ^  j  e^  ic"  ~'^dx~   I  e^  a?" 


DEVELOPMENTS   IN   SERIES.  429 

dx 


_  1    X  [-^n-i  _  ^.^  _  1)  aj"-2+  (7i  —  !)(?!  —  2)a;"-^— &c.]  + 

ig-^  [cc"-^+  {n  -  l)aj"-2  +  (7^  -  l)(;i  -  2) a!"-^  +  &c.]  ; 
consequently,  from  equating  tlie  values  of  5,  we  sliall  have 

^  e^  [ajn-l  _  (^^^  _  1)  ^^n-2  _^  (^^  _   I)  (^  _  2)  «"-3  _  fc]  + 

g-x  ["a^n-i  +  (^  _  1)  ^n_2  _^  ^^  _  1>^  (^^  _  2)  a?"-3  +  &c.]  = 


71  +  1    '    2.3  (/i  + 3)    '    2.3.4.5(^  +  6) 

whicli  needs  no  correction,  supposing  its  members  to  com- 
mence with  X.     If  we  put  n  =  2  and  a?  =  1  in  this  equation, 

we  have    e-^=l:=l  +  ^  +  -^_  +,  &c.; 

which  is  the  same  result  that  Mr.  Young  has  found  at  p. 
100  of  his  "Integral  Calculus,"  from  which  the  example  has 
been  taken. 

5.  To  find  the  generating  function  of  the   series   whose 
•general  term  may  be  expressed  by  the  term 

1 

(j)  +  qn)  (r  +  sn)  &c. ' 

in  which  n  stands  for  the  number  or  place  of  the  term  in 
the  series. 

Because     ^  _^     =  x^  ±  xi       +  ««       ±,  &c., 
1  +  a;  -i-)       ' 

if  we  multiply  the  members  of  this  equation  by  dx  and  take 
the  integrals  of  the  products,  we  shall  have 


480  DEVELOPMENTS   IN   SERIES. 

1       I        XI         .  X'i  ,       X'i  XI  ,       . 

-    /   :; (^^  ~ ±  ?r  -\ TT  ±)  ^^-  5 

consequently,  if  Xi  represents  the  sought  function,  we  shall 
have 

\    r   ^  ^^i        ^+5        ^+3 

Xi  =  -      /    :r-^-r-  dx  =   -—   ±    --r-    H -^   ±,  &C., 

which,  since  its  right  member  vanishes  when  a?  =  0,  we  shall 
suppose  its  members  to  be  so  taken  that  thej  both  commence 
with  £c,  and  extend  to  a?  =  1.     Thus,  if  p  —  0,  we  have 

Xi=— -/  zj = log(l  — a?)  =  -  -f-  — +  — +,&c., 

qJ  1  —  X  q     ^^  ^       q        2q       Zq      '       ' 

when  we  take  1  —  x  for  1  =F  a? ;  and  if  we  put  a?  =  0  in  the 
members  of  this  they  both  vanish,  while  if  we  put  a?  =  1  in 
the  members  they  reduce  to 

i  log  0  =  infinity  =:i(l  +  |  +  |+  &c.), 

a  well-known  result  Again,  if  we  take  1  -\-  x  for  1  =F  aj 
we  shall,  as  before,  by  putting  ^  =  0,  get 

or  log  2  =  1  -  -^  -f  -  _  -  +,  &c. 

Kesuming, 

X 


1  r  -^      -^'      -"'      -+3 

1     I  x<i  ax  _   .T*  x'i  XI         _i_    ft,  ^ 

^ ""  ^  ^  T^x  ~  yvq  ^  j"V^i  "^  Fh^%        ' 


J ii— 1 

then  by  multiplying  its  members  by   a?«     «      dx^   and  in- 

tegrating  by  putting  X^  =  -  J  XiX«     «      dxj  we  get 


DEVELOPMENTS  . 

IN 

SERIES. 

431 

+  2)  C^' 

+  s) 

± 

J- 

+, 

&a 

i.p 

{p 

+ 

2?)  {r  + 

2.) 

Mnltipljing  tlie  members  of  this  by  x^     «       dx^ 

1   r       l^L^i 
and  putting  Xg  =  -   /  Xo  «"     *      o^a? 

and  integrating,  we  shall  have 

-  +2 

and  so  on,  to  any  required  extent. 

1    r        L^E.^1 
Since  Xj  =  -   /  Xi  a? «    ^       dx. 

s  •/ 

when  there  are  but  two  factors  in  the  denominator,  we  get, 
from  substituting  the  value  of  Xj, 

1     r   I_Z_i   ,       /     a?"^       , 

—    /  x"     ^       dx    J dx\ 

qs  J  J    1  ^  X 

which,  integrated  by  parts,  gives 


dx 


x'^  "i       r  XI    .  1  /*  cc»i 

Tr       ~f\    )  1  ^~x  'Tr       p         \     I  l^x 

for  its  value.      Supposing   the  integral  to  be   taken   from 
X  =  0  to  a?  =  1,  we  shall  have 

1    r        H-l-i 
^       -   I  X^x^     1       dx  = 

8   J 

x"     »  r^  XI  dx  1  /         ^*      7  (  ^X 


432  DEVELOPMENTS  IN  SERIES. 

An  example  or  two  may  serve  for  illustration. 

1st.  To  find  the  generating  function  of  the  infinite  series 

_    1  1  1  1         P, 

The  series  is  clearly  satisfied  by  putting^  =  0,  g'  =  1,  and 
r  =  S,s=lj  and  thence  (a/)  becomes 

2d.  To  find  the  generating  function  of  the  infinite  series 

^   _    1  1^1  „ 

^^  -  Li  "  3:6  "^  5.8  -'  *''• 

Since ^  —  —  1,  ^  =  2,  and  r  =  2,  5  =  2,  {a')  gives 

_  1  r'x~^dx  _  1  r\x^dx 
^~~4./ol+a?       4:J  ol-i-x' 

To  find  the  first  of  these  integrals,  we  put  ?/  =  x^,  and  thence 
get  dy  z=  -  x~^dx  —  x  =  y^ 

and  thence  the  first  integral  becomes 

TT  1 

consequently,  ^2  =  j  ~"  7  ^^S  ^  equals  the  generating  func- 
tion of  the  series,  or  X2  =  -r  —  log  4/2,  as  required. 

6.  We   now   propose  to   show   how   to   find   generating 
functions  that  may  be  reduced  to  the  form 

P  +  'l^P  +  'i'I      P  +  H^' 


DEVELOPMENTS   IN  SERIES.  4.S3 

We  shall  assume  the  series 


ax'i  bx'i  cx'i 


±  —cT   +  —- ^±,&C. 


P  +  q        p+2q     '    p+Sq 


"whicli  vanishes  when  a?  =  0,  and  becomes  the  proposed  series 
when  a?  =  1.  Hence,  by  taking  the  differential  of  the  mem- 
bers of  the  assumed  series,  we  shall  have 


—  —  +  1  — +  2 

qds  =  axidx  -^  hx'i       dx  -{-  cx<i       i  dx^  &c., 
whose  integral  being  taken  from  cc  ==:  0  to  a?  ==  1,  gives 

{axi  ±  ^3ci       +  cx'i       ±  &c.)c?a? {b\ 


which  will  clearly  give  the  value  of  the  proposed  series 
when  the  integral  can  be  found. 

Thus,  to  find  the  limiting  function  of 

in  which  1,  2,  3,  &c.,  represent  the  letters  <^,  J,  <?,  &c.,  while 
3,  4,  5,  &c.,  are  represented  hj  p  -{-  q,  p  -\-  2q^  p  -\-  3q,  &c., 
it  is  manifest  that  we  must  have  j9  =  2  and  q=^lj  since  —  is 

p 
used  for  ±  in  the  example ;  and  that  a?^  may  be  moved  with- 
out the  parenthesis,  we  shall  have,  from  (h), 


qs 


{a—  hx  -\-  cx^  —  dx^  +  &c.)  dxi  dx^ 


which  the  substitution  of  the  preceding  values  of  a,  5,  c, 


&c.,  reduces  to  *  —  /   '^^- — ^  ^*'* 


0(1  ■\-'xf 


434  DEVELOPMENTS   IN  SERIES. 

By  taking  tlie  general  integi-al,  we  have 

*  =  (r-21og(l-ha^)-^  +  0, 

which,  being  taken  from  a?  =  0  to  a?  =  1,  gives  5  =  4  —  2  log  2 
nearly,  for  the  generating  function  of  the  proposed  series. 

7.  We  now  propose  to  show  how  to  find  the  generating 
function  of  a  series  of  the  form 

+  7-r-r-F-r— 5  ±,  &c. 


~~  O  +  ^)  ra        {p  +  2q)  m^        {p  +  3^)  m 
Assuming 

^+1  ^+2  ^+3 

*  -  {p  +  q)  m  "^  (^T2q)^'  "^  (p  +  Sq)m'  '^'        ' 
then,  as  before,  we  shall  have 

ods  =  —  dx  ±  — r-  dx  H r-  o?aj  ±  &c. 

/I     .     ^     ■     ^^    ,    c     \   -^  ^^       ^ 

=  ( —  ±  — 5  H ^  ±  &c.  )a?«aa7  =  — — ~  dx, 

\m        m^       m'  I  m  =F  a? 

whose  general  integral  is 

r  p 

I    x^d.r, 
consequently,  we  shall  have 

S  ~  -     I      —:!=r-  dx .(c). 

Thus,  to  fmd  the  generating  function  of 

_    1  1     ^  2-_    ^ 

*  -  2.2  ""  3.4  "^  4.8     '    ""• 

Here,  p  and  q  are  each  1,  and  m,  mr^  m*,  &c.,  are  2,  4,  and  8, 
&c. ;  consequently,  since  we  must  clearly  use  +  for  =F ,  we  have 


lorgxa's  method  of  series.  435 

qs  =    I d.c,     and  tlience     qs  =  x  —  2  log  (2  +  a?)  +  0 

•^     ^'  ~\~  CO 

is  the  general  integral,  whicli,  taken  between  tlie  preceding 
limits,  gives 

s  =  -   C-J^-  dx  =  l  +  'i  4/2  -  2  4/3, 
q  J  q2-{-x  ■  *  ^    ' 

as  required. 

8.  To  illustrate  wliat  is  sometimes  called  Lorgna's  metliod 
of  series,  we  will  apply  it  to  one  or  two  examples. 

1st.  To  find  the  generating  function  of  the  infinite  series 

^~1.2       2.3  "^  3.4      ' 
Because  we  have 

by  multiplying  the  members  of  this  by  dx  and  integrating, 
we  have 
X.&  =  X,  =  fdx  f^^  =  ^^  _  ^  +  |1  _,  &c. 

By  taking  the  integral  by  parts,  we  have 

/dx   I =  X  \o^  (1  +  a?)  —    / dx 

=  {x  +  l)log(l  +  x)  -  X] 
consequently,  we  have 

2  3  4 

{x  +  1)  l<^g  (1  +  ^)  -  ^  ==  ^  -  2~3  -^  I4  -'  ^^•' 

which  needs  no  correction,  supposing  the  integrals  to  begin 
with  X ;  and  thence,  by  putting  1  for  a?,  we  have 

21og2-l  =  A__L  +  J__,&c, 

for  the  generating  function  of  the  given  series. 


436  lorgna's  method  of  series. 

2d.  To  find  the  generating  function  of  the  infinite  series 

Proceeding,  as  in  the  preceding  example,  we  have 


f^dxfdxf- 


■\-  x~  1.2.4       2.3.5    '    3.4.6 
From  the  last  example  we  have 
dx 


— ,  &c. 


A/f 


:(«   +    !)  log  (1    +  2!)  -  X, 


and  thence  we  shall  have 

/  xdx  I  dx  I =  I  x{x  -{-!)  dx  log  {1  +  x)—  I  x^  dx; 

which,  integrated  bj  parts,  gives  the  integral 

/x^       ar\  ,      ,,         ,       x'         r/x^       ar\      dx 


and  thence  the  integral  reduces  to 


(-3  +  2-6)  ^^^(^+^)--9--l2  +  6 


1.2.4        2.3.5       3.4.6 

which  needs  no  correction,  supposing  the  integral  to  com- 
mence with  X.     If  we  put  a?  =  1,  we  have 

2  ,      ^       13  1  1  1 

•^  =  3^"°  ^-  30  =  1X4  -  2:3:5  +  3X6  -'  ^^^ 

for  the  generating  function  as  required. 

3d.  To  find  the  generating  function  of  the  infinite  series 

1  1^  113^  1^315^ 

214-  "*"  2U^  ^  21416%8^  "^  2-.4-.6-.8-.  10^  "^'  ■ 


I 


lorgita's  method  of  series.  437 

From  what  is  done  at  p.  387,  we  easily  obtain  the  formula 


/ 


x"^  fix      _  1.3.5 (2;i-l)  TT 

i^{\  —  x-)  ~  "^A^J %i    2  ' 


in  which  n  is  an  arbitrary  positive  integer,  which,  by  putting 
2,  3,  4,  &c.,  successively,  for  /?,  enables  us,  by  representing 
the  generating  function  by  s^  to  write  the  form 


_      \       r      x'^dx  1  r      x^dx 

~  2.3.4  J    ^(1  -^^)  "^  2^^X576  ^    |/(l-a^)  "^ 

1^3'-  r      a^'c/^ 


/ 


-T.    + 


2.3.4.5.6.7.8^    |/(l-a;-2) 

315^ r        x'' 

2.3.4.5.6.7.8.9.10  ./    |/(1  -  a;-)  "*"'       ' 

whose  integrals,  being  taken  from   a?  =  0   to   x  —  1^  will 

equal  the  proposed  series  multiplied  by  ^ . 

By  taking  the  differentials  of  the  members  of  this  equation, 
it  is  immediately  reduced  to  the  form 

T^  ds       ,  „.  _     x^  x^ 

2  d^  ^^^  ~  ^'^  ~  2A4  "^  2^:3X5:6  "*" 

V.n'x^  S\b'x'' 

+  ,&c.; 


2.3.4.5.6.7.8    '    2.3.4.5.6.7.8.9.10 

and  by  differentiating  the  numbers  of  this  equation  three 
times  successively,  regarding  dx  as  invariable,  we  have 

'^V'^  273  +  273175  +  2:311:5^3:7  +  ^V- 


438  lorgna's  method  of  series. 

Since  the  right  member  of  this  equation  (between  the  paren- 
theses) is  the  same  as  sin  ~ '  x,  we  hence  get  the  equation 

whose  integrals  give 

Bj  taking  the  integral  of  the  members  of  this  equation, 
we  have 

and  taking  the  integral  of  this,  we  also  easily  get 
whose  integral  again  taken  becomes 

/,    _  2   /*       x^dx  2  r   cosxdx 

~  rj  2  4/(l-(^  ~  '^J  V(I-*^' 
and  so  on. 

Kemarks. — The  substance  of  the  last  six  pages  has  been 
taken  from  Young's  "Integral  Calculus,"  from  pp.  99  to  111 
inclusive.  Mr.  Y.  shows,  at  p.  108,  in  a  manner  very 
analogous  to  that  used  by  us  in  the  solution  of  our  last 
example,  that  the  generating  function  of  the  series 

3-  3^5^  Z\^\T  ,  ,       8       ^1 


SECTION  Y. 

INTEGRATION   OF   DIFFERENTIAL   EXPRESSIONS  WHICH 
CONTAIN   TWO   OR   MORE   VARIABLES. 

(1.)  A  DIFFERENTIAL  of  a  function  of  two  or  more  varia- 
bles whicli  is  derived  from  the  function  by  taking  its  differ- 
ential, supposing  the  variables  all  to  change,  is  said  to  be 
compltte  or  exact ;  while,  if  the  differential  is  taken  on  the 
supposition  that  the  variables  do  not  all  change  their  values, 
it  is  said  to  be  incomplete^  inexact^  or  2>'-f-Ttial. 

Thus, 

-  ^ —     du       ydx       xdu  -~yi  ^,         ,  y 

ydx  +  xdy  =  dyx,     -^  -  ^-  =  a^  =  "^  ^' 

are  complete  or  exact  differentials,  while  ydx,  xdy  —  ydx^ 
are  incomplete  or  inexact  differentials,  provided  there  is  no 
assigned  relationship  between  x  and  y ;  other  examples  of 
exact  differentials  will  be  obtained  by  reversing  the  exam- 
ples at  pp.  7  to  12. 

(2.)  It  is  easy  to  perceive  that  if  'M.dx  +  ^dy  is  an  exact 
differential  of  two  variables  x  and  y^  that  its  integral  may 
be  found  by  the  following 

RULE. 

1.  Take  the  integral    /  M.dx  on  the  supposition  that  y  is 

•constant  or  invariable,  and  add  to  the  result  the  integral  of 
all  the  terms  in  l^dy  which  are  independant  of  x  or  do  not 


440  DIFFERENTIAL   EXPRESSIONS 

contain  x ;  then  the  result,  increased  bj  an  arbitrary  con- 
stant, will  be  the  complete  or  exact  integral. 

2.  Or  we  may  take  the  integral    /  Nc^y,  on  the  supposition 

of  the  constancy  of  a?,  and  increase  the  result  by  the  integral 
of  that  part  of  Mci.r  which  is  independent  of  y,  and  an  ar- 
bitrary constant  lor  the  same  integral  as  before. 

Bemarks. — It  clearly  results  from  the  rule,  that  when 
Mdx  -f-  '^dj/  is  an  exact  or  complete  differential  of  a  func- 
tion (M  and  N  being  functions  of  x  and  y)  we  must  have 

-7—  =  -7- ;  which  is  called  Euler^s  Criterion  or  Condition 
ay         dx 

of  Integrahility  of  the  differential  McZa?  +  Nc?y  (see  p.  22) 
Hence,  since    /  M<^fe  —    I  Ndy^  we  have 

d   I  Mdx 

dy         -^» 

and  from  -7-  =  -7— ,  we  have  aN  =  -7-   dx.  which  gives 
dx        dy  ^  dy       ^  ° 

N  =    I  -r-  dx\  consequently,  we  must  have 


dy 

d  I  Mdx 


rdU  , 


dy  J   dy 

which   is  agreeable    to   Leibnitz's   rule   for  differentiating 

under  the  sign    /  ;  noticing,  that  the  right  member  of  this 

equation  is  independent  of  the  Urst  integral,   or  that  with 
respect  to  x. 

(3.)  To  illustrate  the  rule,  take  the  following 


with  two  01^  more  variables.  441 

e:xamples. 

1.  To  find  the  integral  of  {Qxy  -  tf)  dx  +  (S.^^  -  2xij)  dy. 

Since  x  enters  into  every  term  of  the  coefficient  of  dy^  it 
is  clear,  if  the  proposed  differential  is  exact,  it  will  be  suffi- 
cient to  find  the  integral  /  (Qxy  —  y")  dx^  supposing  y  con- 
stant ;  consequently,  o.-c-  y  —  y-  x  -\-  C  must  be  the  integral, 
which  is  evidently  true,  since  it  equals  the  integral 

f{U^-2xy)dy,      ' 

regarding  x  as  constant. 

Remark. — ^Because,  in  this  example,  M  and  N  are  repre- 

dM. 

sented  by  Qxy  —  ?/"  and  3a?^   -  2s?  i/,  which  give  -j-  =6x  —  2y 

,  f/N      ,        ^      ,        .     .        ^  .  ,  .,.      ^M      c/N 

and  -J-  =:  tx  —  2y,  the  criterion  oi  Integra  bility,  -j~  —  -y- , 
cix  dy        cix 

is  satisfied. 

2.  To  find  the  integral  of  i^x''  +  2axy)  dx  +  {ax''  +  Sy^)  dy. 

Here  /  (3.''  +  2axy)  dx  =  x^  +  ax^y, 

to  which  adding  the  integral  of  Sy'dy,  the  part  of(ax''-{-Sy^)dy 
which  is  independent  of  a?,  and  we  have  ar^  +  ax'y  -\-  y^  +  C, 
after  adding  the  constant  C,  for  the  exact  integral. 
The  same  integral  is  also  found  from  the  integral 

Jiac^  +  3y-)  dy  =  axy  +  y\ 

by  adding  the  integral   3  /  a?'-^  dx  ~  x^  +  G,  the  integral  of  the 

part  of    (3a3^  +  2axy)  dy    which  is  independent  of  y,  to  it 
The  criterion  is  also  satisfied. 

19* 


412  DIFFERENTIAL   EXPRESSIONS 

3.  To  fmd  the  integral  of  ^^^  =  -J^"-, 


Here  the  integral 


^  +  y'     ""  V'  +  '/r      y-  +  ss^ 
d.v 


/^-^^    r_J^_^  =  tan-^  +  C; 
^  y'  +  X-     J  x^  y 


y 

and  the  integral 

dy 

f--^±y-^=     ri^Zl..tan-?  +  C, 
J       y^  ^-  9r      J       ^  ^x^  y 

y- 

the  same  as  before. 

4.  To  find  the  int-cgral  of  —jr-r- — s;  +  V^V' 

Here  M  =  —rr-^ -^r  aiid  N  =  y.  which,  since  they  do  not 

contain  y  and  a?,  give  -j-  =  0  and  y-  =  0,  which,  being 

naught,  may  be  regarded  as  satisfying  the  criterion  of  in- 
tegrability. 

Hence,  the  proposed  differential  maybe  regarded  as  having 
an  exact  integral,  which  is  also  evident  from  principles  here- 
tofore given,  since  each  term  of  the  proposed  dilFerential  is 
clearly  the  function  of  a  single  variable.    Indeed,  the  integral 

r       x'dx        __  X  j/  (cz-  4-  X-)  _  cr    r         d.c 
J    j^{a^  +  aj2)  ~"  'I  '2  J    4/(rrT" 

log  {x  +  \^X'  +  (/-) , 

and  since  /  ydy  =  ~ ,  the  integral  of  the  proposed  differ- 
ential is  of  course  found,  after  the  addition  of  the  arbitrary 
constant. 


i/ia"  +  x')  2  '2  J    ^{rr  -^x') 

X  {/{a^  +  X-)       a- 
2  2 


WITH  TWO   OR   MORE   VARIABLES.'  443 

5.  To  find  the  integral  of 

i^ay?  -{-  ^dcfy)  dx  +  hMy, 
Here  we  have 

fudx  =    [{^aa?  +  2lxy)  dx  —  ax"  ■\-  hx^y  +  Y, 

in  whichi  Y  stands  for  the  arbitrary  constant  in  tlie  inte- 
gration with  regard  to  x  while  it  may  be  a  function  of  y^ 
since  y  has  been  supposed  to  be  constant  in  the  integration 
with  reference  to  x. 

To  determine  Y,  we  take  the  differential  of  the  preceding 
equation,  regarding  x  as  constant,  and  thence  get 


d  I  M-dx        j^  r.  d  I  M.dx 

consequently,  since 

d   I  U.dx 

1^  —  haP    and j =  haP. 

dy 

d   I  Mdx 

we  have  N -r ==  0. 

dy 


Hence,  the  sought  integral  is  reduced  to 

dy      I    ^ 
which  might  also  liave  been  expressed  by  the  form 

J  Ndy  +  y  (M -—-^ j  dx  =bx'y  +  «a^  +  C, 

the  same  as  before. 


444  DIFKEKEXriAL   EXPliESSIONS. 

Remark:. — We  Lave  performed  this  solution  according  to 
the  common  metliods,  in  order  to  show  that  they  are  sub- 
stiiiitv'!  ;y  tiic  s:une  as  our  rule. 

(4.)  It  is  easy  to  perceive  that  our  rule  may  be  extended 
to  lind  the  integral  of  a  differential  consisting  of  any  number 
of  terms,  like  ^Ldx  +  l^dij  +  P^^^  +,  &c.,  by  adding  to  the 

integral    /  Isldx  taken  relatively  to  a?,  the  integral  of  all  the 

terms  in  l^dy  which  are  independent  of  a?,  and  then  adding 
the  integral  of  all  the  terms  of  Ydz  that  are  independent  of 
either  x  or  y  (or  both  of  them),  and  so  on  to  any  required 
extent.     Thus,  the  integral  of 

ydx  _^  (x  +  2ay)  dy  _  {xy  +  af)  ^^ 

z  z  z- 

/ydx       y    r  -,         V^ 
- —  =  -   I  dx  =  ^—, 
z         z  J  z 

f/2.Wy  =  «-f,  and  („y  +  a,^)/-J  =  ?^^, 
whose  sum,  corrected  by  the  addition  of  an  arbitrary  con- 

XtJ  -f"  O-  XT 

stant,  is  ' ~  +  0,  which  expresses  the  integral  as  re- 

z 

quired. 

If  ^Idx  H-  NcZ?/  +  Ydz  +,  &c.,  is  the  differential  of  some 
function  of  .r,  y,  2,  &c.,  of  u^  we  shall  have 

^g       du      T,-r       du      _,       du     . 

which  give 

c?M        d-u        c/N        c^?/        ^M        d"-u 


dy        dxdy^     dx        dydx^      dz        dxdz'' 
,      dV        d}u       dN        dhi       dF        d'u      „ 
dx       azdx      dz        dydz '    dy        dzdy 


WITH   TWO   OR   MORE   VARIABLES.  445 


Because        -, — -  =  -7-7- , 
daxly       dydx 

and  so  on  (see  p.  22),  we  shall  have 

^M  _  d^      d}l 
dy  ~   d^  '      d3 


d/u 

d'^o 

djidz 

dzdji' 

I  have 

d^ 

dji  ' 

cZN 
"dz 

dY 
-dy 

kc, 


for  the  Criteria  of  Inteyrahility  of  a  diflerential  of  the  pre- 
ceding form ;  which,  being  supposed  to  contain  n  different 

variables,  will  give  — -  equations,  like  the  preceding, 

in  the  criteria  of  integrability,  since  by  the  known  principles 

of  combinations,  — -  shows  how  often  two  may  be 

taken  out  of  n  different  things. 

It  is  hence  clear  that  any  differential  which  satisfies  all  the 

71  ( ?h     T  I 

-  ^      - — -  criteria,  can  be  integrated  by  the  preceding  method, 

and  its  integral  wall  be  exact ;  but  if  the  criteria  are  not  all 
satisfied,  the  integral  can  not  be  found,  and  must  be  incom- 
plete or  inexact;  hence  the  importance  of  examining  the 
conditions  of  integrability  before  we  proceed  to  integTate  the 
equation,  becomes  too  evident  to  require  any  further  notice. 

(5.)  Supposing  Adx  +  Bdy  +  Cds  +,  &c.,  to  be  an  exact 
differential,  or  one  that  satisfies  all  the  criteria  of  integra- 
bility, and,  at  the  same  time,  suppose  each  of  its  coefficients, 
A,  B,  C,  &c.,  to  be  of  71  dimensions  in  terms  of  its  variables, 
a?,  y,  2,  &c.,  or,  which  is  the  same,  suppose  the  equation  to  be 
homogeneous,  the  degree  of  homogeneity  being  71 ;  then,  we 

,     ,        ,1    , .,    •   ,        1        A;??  +  By  +  Cz  +  &c. 

propose  to  show  that  its  integral  — , 

...  n  +  1  ' 

provided  n  is  different  from  —  1. 

Thus,  since  y,  2,  &c.,  may  clearly  be  expressed  by  xy\  X3\ 


446  DIFFERENTIAL  EXPRESSIONS 

&C.,  because  the  differential  may  evidently  be  supposed  to 
have  been  obtained  by  regarding  x  alone  as  variable,  it  must 
be  expressed  by  the  form  K.dx  -h  V»y'dx  +  (^z'dx  +,  &c. 
Because,  from  the  nature  of  homogeneity,  each  term  of  this 
differential  must  be  supposed  to  contain  the  factor  x""  dx^ 


X 


n  +1 


which,  integrated  by  the  rule  at  p.  254,  gives for  the 

common  variable  factor  of  the  terms  of  the  integral ;  conse- 
quently, the  integral  must  evidently  be  expressed  by 

Ax  +  Bxy'  +  Oxz'  +  &c. 


n  -\-  1 


+  const, 


or  its  equivalent, ; h  C, 

^  n  -\-  I 

C  being  the  constant. 
It  may  be  noticed  that  if  /i  =  —  1,  the  integral 


I  x'^dx  =    I  —  =  log  X  ; 


consequently,  when  n  =:  ~  1,  it  results  that  log  x  must  be 
a  factor  of  the  integral  of 

Adx  -f-  By'dx  -f  Cz'dz  +,  &c. 

Hence,  when  n,  called  the  index  of  homo'jeiieity^  is  dif- 
ferent from  —  1,  change  dx^  dy^  dz,  &c.,  severally  into 
X,  y,  2,  &c.,  in  the  differential  Adx  ■{■  Bdy  +  Cdz  +,  &c., 
divide  the  result  by  the  index  of  homogeneity,  increased  by 
unity,  and  add  an  arbitrary  constant  to  the  quotient  for  the 
integral 

EXAMPLES. 

1.  To  find  the  integral  of  {Sx^  +  2a.Ty)  dx  +  {ax^  +  3y-)  d,j. 

Here  the  index  of  homogeneity  is  clearly  2,  being  the  sura 

of  the  indices  of  x  and  y  in  each  term  of  the  differential ; 


WITH   TWO   OR   MOllE    VARIABLES.  447 

consequently,  since  the  differential  is  clearly  integrable,  by 
changing  the  differentials  dx  and  Jy  into  x  and  y,  we  have 
(3.-t''^  +  2aa?y)  x  -f-  {ax^  4-  3y-)  y.  Performing  the  requisite 
multiplications,  and  uniting  like  terms  of  the  products,  we 
have  Zx^  -f-  Zaxhj  +  3y^,  which,  divided  by  2  +  1  —  3,  gives 

Zx^  +  Zax-y  +  3y'  , 

o 

and  adding  the  constant  C  to  this,  we  have  x^-^ax^y-\-y'-\-0 
for  the  integral  of  the  proposed  differential, 

2.  To  integrate  {^x'-\-1lxy-^]r)dx^  {hx^—^xy-V  ^g/)  dy. 
This  being  both  integrable  and  homogeneous,  we  have,  as 

before, 

Zx^  +  Ihx-y  —  Zxy-  +  hxhj  —  Qxy''  +  Zcf       ^ 

^— —  +  C  := 

x"  +  hxhj  -  Zxf  -f  cy^  +  C 
for  the  integral. 

3.  To  integrate  (2/^  4-  Zf)  dx  +  i^x-y  +  9.^y-  -f-  8/)  dy. 
The  answer  is        yV  +  3y''^  +  2?/'*  +  C. 

4.  To  mtegrate  -^ f- ~—-  +  — — t^ —  . 

°  z  z  z^ 

Since  the  indices  of  x  and  y  are  positive,  while  those  of  ^, 
in  the  denominators,  are  to  be  considered  as  negative,  it  is 
manifest  that  the  index  of  homogeneity  is  naught.  Hence, 
it  is  easy  to  perceive  that  the  integral  is  expressed  by 

''jLzJ:  +  c. 

z 
6.  To  integrate  the  integrable  and  homogeneous  differential 
dx  I  ^  X        \  dy 


+     1 


^j  putting  y  —  xy\  the  differential  is  readily  reduced  to 


4AS  DIFFERENTIAL   EXPRESSIONS 

dx  dc  du)  _  dx 

whose  integral  may  be  expressed  by 

log  X  +  log  C  =  log  Cx. 

If  C  =  C  [1  +  V  (1  +  y'%  we  have 
log  C^'  =  log  C'  [x  +  X  |,/(1  +  2/'^)]  =  log  C  [.r  +  |/ (0.^+2/=)], 
wbicb  is  tlie  well-known  form  of  the  integral  as  determined 
by  the  ordinary  process  of  integration ;  noticing,  that  the 
integral  appears  under  quite  an  undetermined  form,  on 
account  of  the  terms  that  have  destroyed  each  other,  agree- 
ably to  what  is  said  at  pp.  445  and  446,  the  index  of  homo- 
geneity in  this  example  being  —  1. 

6.  To  integrate  the  integrable  and  homogeneous  differential 

xdy  ydx. 

x^  +  y^        ar  +  2/^  * 

Here,  the  index  of  homogeneity  is  —  1,  and  the  differ- 
ential is  readily  reduced  to 

y'dx  y'dx 

xiyVy^')  ~  ^i  +  y'') ' 

whose  terms  destroy  each  other,  and  have  the  differential 

dx 

— -  for  a  common  factor;  consequently,  it  is  clear  that  the 

X 

integral  is  here  under  a  more  undetermined  form  than  in  the 
preceding  example.  It  is  hence  clear  that  such  integrals  as 
these  ought  to  be  avoided  as  much  as  possible. 

(6.)  We  will  now  show  how,  according  to  the  preceding 
principles,  to  integrate  a  differential  expression  of  the  form 

Q^dx^  +  'Rd.cdy  +  ^df, 

in  which  dx  and  dy  are  supposed  to  be  constant  or  invariable, 
and  X  and  y  are  regarded  as  independent  variables  ;  then,  be- 


WITH  TWO    OR   MORE    VARIABLES.  449 

cause  eacli  term  of  the  expression  contains  two  dimensions 
of  the  differentials,  it  is  said  to  be  of  the  second  order  of 
differentials,  or  of  the  same  order  as  that  of  the  differential 
of  a  differential,  and  in  dimensions  the  expression  is  said  to 
be  of  the  second  degree,  or  of  two  dimensions.  (See  Lacroix's 
"  Calcul  IntegraV  p.  232.) 
It  is  easy  to  perceive  that  we  may  consider  Q^dx^  and  S  ///- 

to  have  been  derived  from    /  Q^a?  and   /  Sdy  by  taking  their 

differentials,  regarding  x  and  y  as  separately  variable  in  the 
expressions ;  consequently,  the  proposed  differential  may  be 
supposed  to  have  been  obtained  from  taking  the  differentials 

of  the  differential  dx   j  Qdx  ■\-  dy  I  Sdy,  on  the  supposition 

of  the  constancy  of  dx  and  dy^  while  the  first  integral  is 
taken  on  the  supposition  of  the  constancy  of  y,  and  the 
second  supposing  x  to  be  constant. 

Hence,  by  taking  the  differential  of  this  assumed  expres- 
sion by  considering  x  and  y  both  to  vary,  and  by  diffei'en- 

tiating  under  the  sign    /  ,  according  to  the  rale  of  Leibnitz, 

given  on  page  440,  we  shall  have 

Qdx^  +  dxdy   I  -j-'  dx  ~^  dydx  I  —  dy  +  Sdy^ ; 
which  must  clearly  be  identical  with  the  proposed  differen- 
tial, and  thence     I  -^  dx  -{-   I  -j-  dy  =  B,: 

Differentiating  the  members  of  this  equation  with  regard 
to  X  alone  as  variable,  and  differentiating  the  second  term 
under  the  sign    /  ,  by  the  rule  of  Leibnitz,  we  shall  have 

^dx  +  dxj^^dy=^,dx, 


450  DIFFERENTIAL  EXPRESSIONS 

or  we  nave  -7-^  +    /  -rr  dy  =  -y-: 

ay        J  dor     ^         dx 

and  removing  the  sign    /  ,  by  differentiating  the  members 

of  this,  regarding  y  alone  as  variable,  we  have 

drq       d'S       ^R 


dy'^        dj?        dxdy ' 
r7-R 

which  is  the  same  as  -7-—  for  the  condition  of  integrabilitj 

of  the  proposed  differential. 

(7.)  We  now  propose  to  show  how  to  find  the  integj*al  of 
a  differential  expression  of  the  form  Vd-y  -\-  Qdr,  given  by 
Lacroix  at  p.  234  of  his  work,  in  which  x  is  the  independent 
variable,  and  y  is  regarded  as  being  a  function  of  a*,  and 
P  and  Q  are  supposed  to  be  functions  of  x,  y,  dx,  dy. 

Putting  dy  =  i^dx,  and  taking  their  differentials,  regarding 
dx  as  being  invariable,  which  we  clearly  may  do,  we  have 
d'^y  =  dj)dx;  which,  substituted  for  d^y,  reduces  the  given 
differential  to  the  form  {Fdj?  +  Qdx)  dx,  which  may  evi- 
dently, as  in  Lacroix,  be  represented  by  the  more  general 
form  (M.dj?  -\-  No?a?)  dx",  whose  integral  ought  evidently  to 

be  of  the  form  udx-^ ;  or  we  must  have  icz=  I  Mdp  +  Y,  sup- 
posing the  integral  to  be  taken  with  reference  to^,  regarding 
x  and  y  as  being  constants,  and  Y  as  being  a  function  of 
them.  Differentiating  the  members  of  this  equation,  regard- 
ing M  as  being  a  function  of  x  and  y,  observing  the  rule  of 

Leibnitz  for  differentiating  under  the  sign   /  ,  we  shall  lu 


ave 


,    rdu  ,      ,    rdu  ,      dv  ,     dv  , 

du  =  mip  +  dxj   ^  dj>  +  dyj   ^  dj>-^  ^  dx+  ^  r/y, 
which,  compared  to  Mdj?  +  'Ndx,  gives 


WITH  TWO   OR   MORE   VARIABLES.  451 

^^       rd^i  ^  rdu  ,       dY      dV 

which  must  be  an  identical  equation. 

To  remove  the  sign  /  ,  we  differentiate  this  twice  suc- 
cessively with  reference  to  ^,  and  thence,  since  V  does  not 
contain  J?,  get 

d^  __d}l        rdU  ^        dU         dV 
dp  ~  dx        J   dif    ^        dy  ^       dy  ' 

,  cV-^        d}\i       ^d\l        d-W. 

and  -j-r  —  -j—r  +  2  -,-    +  -^-j-  p (1), 

dp-        dxdp  dy        dydp  ^ 

an  equation  freed  from  V,  that  must  be  satisfied.     Hence, 

dy  ~  dp        dx        dy  ^        J  dy    ^' 

,   dY      _^      fZN         ^M         dVi.    ,        f ^M  , 
^"^^   Tx--^-^  -dp^^  -dx^-^  -dy^^'  J  IFx^^'^ 

and  since  the  dilferential  of  the  first  of  these  with  reference 
to  X  equals  that  of  the  second  with  reference  to  y^  we  have 

dy       dpdx      dpdy^^  djT  '^'-'dxdy^'^  dif  ^'''^  ":^''^' 
When  a  proposed  differential  satisfies  (1)  and  (2),  bj  sub- 
stituting the  values  of  -r-  and  -7-  in 
dx  dy 

du^^dp^dxj-^^  dp  +  dyj  -^  dp^-  ^  ^+  ^  %, 
we  shall  get 

du  =.  Mdp  +  (^  -        ^  +  _  ^  +  ^-^j  ax  + 


dp  -^        dx^        dy 

dU  _  dU 
dp        dx        dy 


/dN  _  dU  _  dU    \ 
\dp        dx        dy  -^l    '^' 


452  DIFFERENTIAL   EXPRESSIONS 

which  is  freed  from    /   and  under  the  form  of  a  differential 

of  p^  X,  and  y,  whose  integral  can  clearly  be  found  by  the 
rule  in  (4),  at  p.  444. 

Thus,  to  find  the  integral  of 

(^.vydy  +  a^ydx)  d/y  +  xd]f  +  (y  +  a?^)  dy'^djx  + 

(2  +  3y)  xydyd^  +  y^d'j^ 

from  what  has  been  done,  we  put  jpdx  and  djpdx  for  dy  and 
d^y^  and  thence  get 

M  =  2.^yj?  +  x-y,  N  =  £cy  +  (y  +  d?)f^  (2  -\-.Zy)  xyp  +  y', 

which  give 

_=6^  +  2(y  +  .^)^^  =  2y-^ 

=  ^^^  +  ^•^'11^  =  2^^' 

which  will  satisfy  (2) ;  consequently,  the  expression  is  an 
exact  differential,  which  is  reducible  to  the  form 

du  =  i^xyp  +  x^y)  d^j  +  {yp^  f  ^typ  +  y^  dx  -\- 

{xp/  H-  x^p  +  Sxy^)  d-y. 

The  integrals  of  the  first  term  of  this  relative  top,  and  those 
of  the  two  last  terms  relative  to  x  and  ?/,  by  omitting  the 
terms  containing  p  in  them,  when  added,,  give 

xyp^  +  x^yp  +  xy^  +  C 

for  the  value  of  'w ;  consequentlj^,  since  the  sought  integral 
evidently  has  the  integral  =:  udj?^  we  shall  have 

udn^  r=  xydy^  +  xnjdydx  +  xifdj?  +  Qdd^ 

'for  the  required  integral.  It  is  easy  to  see  that  we  may,  in 
much  the  same  way,  proceed  to  determine  the  integral  of  any 


WITH   TAVO   OR   MORE   VARIABLES.  453 

diiTerential  expression  between  x  and  ?/,  when  it  is  of  any 
order  of  differentials  greater  than  the  first. 

Concluding  Eemarks. — Because  the  differential 

dx 

A  {r  —  X)  dx  A  7  (^"  ~~  ^^''^  "^  ^")^ 

{r'-2rx+¥f  ~  ^'' 

{f-  —  2r.'»+  Ir')    ^rdx 
— —  —  ^(t 


dr 
we  thence  get 
A     r      ir-x)dx       __        .^    r{7»-2rx  +  'b^)~^'rdx 

^  {f  -  2nx  +  &n^  "  1 L   

dr 

(/■^  —  2ra;  +  Vf 

=  Ad ^ +  0 

dr 

for  the  integral. 

It  is  hence  easy  to  perceive  how  the  forms  of  differentials 
may  be  sometimes  changed,  so  as  greatly  to  facilitate  their 
integration,  by  taking  the  differential  of  them  with  refer- 
ence to  a  constant  in  them. 


SECTION  VI. 

INTEGRATION   OF   DIFFERENTIAL   EQUATIONS   OF  THE   FIRST 
OKDKR   AND   DEGREE,    BETWEEN   TWO    VARIABLES. 

(1.)  It  is  manifest  that  a  differential  equation  between 
any  n\imber  of  variables,  when  the  variables  are  separated 
from  each  other,  is  such  that  the  integral  can  always  be 
found ;  and  if  the  terms  of  an  equation  are  of  an  integrable 
form,  it  may  evidently  be  integrated  by  the  methods  given 
in  Section  Y. 

(2.)  If  we  have  an  equation  of  the  form  ^idy  +  Ydx  =  0, 
between  x  and  y,  such  that  X  is  a  function  of  x  alone,  and 
Y  a  function  of  y  alone,  then,  dividing  the  equation  by  XY 
the  product  of  the  differential  coefficients,  it  is  reduced  to 

ut/       dx 

=^  +  :^  =  0,  which  is  clearly  an  integrable  form,  or  such 
X         X 

that  the  integral    /  ^-  +    I  --  =  0  can  be  found. 

Thus,  the  particular  differential  equation 
{x  +  Ifdy  =  (y  -f  Ifdx 
dy  dx 


is  reducible  to 


{y  +  1)^        {x  +  1)' 


2> 


whose  integral  is   ^r-, -— ,  =  zr  +  0. 

2  (y  +  1)-       X  +  1 

(3.)  Similarly,  the  differential  form 

XY^y  -I-  X,Y,d^  =  0, 


EQUATIONS   OF  THE   FIRST   ORDER.  455 

divided  by  the  partial  product  XY,  of  the  differential  coeffi- 
cients, becomes  ■   +  ^  -L  '■  =i  0, 
Yj  A. 

in  which  the  variables  are  separated,  and  it  is  clearly  an 
integrable  form,  or  such  that  the  integral 

can  be  found. 

Thus,  the  particular  differential  equation 

{x  +  l)i/dx  -  {y"'  +  \)xdy  =  0, 
divided  by  xy"^^  becomes 


(i  +  l)^,  =  (i  +  y,y. 


which  is  clearly  an  integrable  form,  the  integral  being 

X  +  log  X  ^=  y h  C. 

J 

(4.)  The  equation     dy  +  Vydx  —  Qdx, 

sometimes  called  a  linear  eqxiatio7i^  can  have  its  variables 
separated  by  assuming 

dy  +  Yydx  ■=■  0,     which  gives     -^  r=  —  Ydx^ 
whose  integral  may  evidently  be  expressed  by 

log  2/  -  log  C  =  log  I  =  -  Jvdx, 

or  using  e  for  the  hyperbolic  base,  y  =  Qe~f^'^^.  To  adapt 
this  to  the  proposed  question,  we  may  suppose  0  to  vary ; 
consequently,  by  taking  the  differential  of  y  on  this  suppo- 
sition, we  sball  have 

^y  =  -  Qe-P^^dx  +  dQe-A'^^ 


456  EQUATIONS   OF  THE   FIRST   ORDER 

Bj  substituting  y  and  dy  in  the  proposed  equation,  and 
erasing  the  terms  that  destroy  each  other,  we  have 

dQe-P^^  ^  ^dx,    or    dQ  =  eP^'(idx, 

whose  integral  gives  C  =    /  ef^^'^-'Q^dx  +  C.     Hence,  from 
the  substitution  of  this  value  of  C  in  that  of  ^,  it  becomes 

y  =  e-/^'^(feA^'Qdx  +  C') 

for  the  integral  of  the  proposed  equation. 

Remarks. — Hence,  the  integral  obtained  from  a  very  sim- 
ple case  of  the  proposed  diiferential  equation,  by  the  varia- 
tion of  the  arbitrary  constant,  has  enabled  us  to  find  the 
integral  when  taken  in  its  utmost  extension. 

OtJierwise, — By  assuming  y  =  Xs,  we  shall  get 

dy  =  zd'^  -f  Xc?s, 

which  values  of  y  and  dy^  substituted  in  the  proposed  equa- 
tion, reduce  it  to 

zd^  +  Xo?0  -h  VXzdx  =  Qo?a?, 

in  which  X  being  arbitrary,  we  may  assume 

dz  -f  Vzdx  =  0     or    z  =  e-P'^'', 
and  thence  get 

^  ^  Qd»  ^  ^-^Q.7aj  =  ef^'^^Q^dx, 

z 

whose  integral  is  X  =    Cef^'^''Q,dx  +  C. 

Hence,  from  the  substitution  of  these  values  of  X  and  ^, 
we  shall  have 

y  =:   X,2   r=   6-/1'^-  (feA'^Qlx  +  c) 

for  the  integral,  the  same  as  found  by  the  preceding  method. 


B:r:wE:-:N  two  variables.  457 

Eemark. — This   method  of  integration  has  been   taken 
from  p.  254  of  Lacroix's  "Calcal  Integral." 

(5.)  The  more  general  differential  equation 

dy  +  ^ydx  =  Qy'^  + '  dx 

can  readily  be  reduced  to  the  preceding  form. 

For  by  multiplying  its  terms  by ^^-^^ ,  it  becomes 

ndy        nVdx  ^  , 

-which,  by  putting  ^  =  —  ,  becomes 

dz  —  riFzdx  =  —  7iQdXj 

whicb  is  of  like  form  to  the  differential  equation  in  (4). 
Hence,  by  putting  —  nP  and  —  71  Q  for  P  and  Q  in  the 
integral  in  (4),  we  shall  have 


n/Fdx  /_  ^    A-"/p<^^  Qdx  +cA 


for  the  integral  of  the  preceding  equation,  and  thence  we 
get  y.     (See  p.  192  of  Young's  ''Integral  Calculus.") 

To  illustrate  the  preceding  formulas,  take  the  following 

EXAMPLES. 
1.  To  find  the  integral  of  dy  +  ydx  =  ax^  dx. 
Comparing  the  equation  to  that  in  (4),  we  have 

P  =  1     and     Q  =  ax^^     and  thence      /  Vdx  =  a?, 
which  reduces  e/^"^^  to  e'',  and 

feA^^  Qdx  +  C    reduces  to     afe^afdx  +  C  ; 


458  EQUATIONS   OF  THE   FIRST   ORDER 

whose  integral,  being  found  by  integrating  by  parts,  gives 
a  fe'a^dx  +  C  =  ae"  {x"  -  2x  +  2)  +  0'. 

Hence,  from  y  =  eS''^^  (Jef^^^  Qdx  +  c)  , 

we  get  y  =  a  {x"  —  2a?  +  2)  -f  C'e-' 

for  the  sought  integral. 

2.  To  find  the  integral  of    dy  +  ydx  =  aa?"  dx. 
Here  we  have  P  =  1,   Q  =  «a?",   and  thence 

e/^^"^  =  e'-  feA^""  Qdx  +  C  =  a   fe^x'^dx  +  C, 

which,  integrated  by  parts,  as  before,  becomes 

e'a  [a?"  —  nx''-'^  +  n  (ti  —  1)  aj"-^  —  &;c.]  +  C; 
consequently,  we  shall  have 
y  =  e-A'-  {eP^^  Qdx  +  CO 
=  Q  [a?"  —  Tia;"-^  +  n  {n  —  1)  x""-^  —  &c.]  +  C'e"* 
for  the  required  integral. 

xdx  CLX 

3.  To  find  the  integral  oi  dy  -{-  y ^  =  ^  dx. 

Here  we  have 

^~  l+aj^'    ^~  1  +  a^' 
TPc/aj  =  log  (1  +  aj-)^    ^/P'^^  =  ^logvd  +  x')^ 

and      y^A'^^  Qria?  +  C^  =  Q  fe''^^^^'  ^  -'>  ^^^  +  C 

=  Q^iogva  +  x')  +  C'; 
consequently, 

y  =  e-A^-'x  IfeA^^Qdx  +  c)  ==  Q  +  0^6-^^^^^^  +  ^ 
is  the  required  integral. 


BETWEEN   TWO   VARIABLES.  459 

4.  To  find  the  integral  of  di/  -{-  ydx  =  y^  xdx. 
Here,  from  the  formula  in  (5),  we  shall  have 

and      J~  ne-^f^^-  Qdx  +  C^  =J*-  2e  - '-  xdx  +  0' 

=  .--  {x  +  l^+0^. 
consequently,  we  shall  have 

'  =  ?  =  (^  +  '^)  +  CV' 
for  the  required  integral. 

5.  To  find  the  integral  of  dy  -f-  7/---—  -  ,/^^^^.. 

thence  we  have 
y  Pdx  =  —  log  ^/(l  _  x'')     and     6'^/'^'^^  =  e-i^g  i/a-*«)^ 
Hence,  we  shall  liave 

which  gives       z  ~  -  =  1  +  C'  6^°2^^^-^^> 

for  the  right  integral. 

6.  To  find  the  integral  of  dy  -   ^^^  -  __^^ 

^        1  +  a^'        l+ar'- 

Here  '  Y  =  ~  _^_       o  — —-. 

yp^^=-iog|/(i+a^), 

J  ~7(r-r^)'     ^»d  thence 


460  EQUATIONS   OF  THE   FIRST  ORDER 

consequently,  y  =  ax-\-G'Vl  +  x^  is  the  integral. 

7.  To  find  the  integral  of  dv  +  ^^-  =  y^xdx. 
°  ^      1  —  a?-       ^ 

Here  p  =  _^_,    Q  =  a!,     n  =  - -, 

fWx  =  —  log  (1  -  x')^,     n  fvdx  =  log  (1  -  x')\ 

and  ^V^*^^  =  e '«« ^'  -  ^'^*^  =  (1  -  x'')^i 

from  the  nature  of  numbers  and  their  hyperbolic  logarithms. 
We  also  have 

consequently,  from    s  =  — ^  r=  — ^-j^  =  2/% 

2/        y    ■' 

since  s  =  ^"A'^^  (—  w  J e-''f^^^  Q,dx  +  C) 

we  shall  have  y^  =:  C  (1  -  ar^)^  -  ^  (1  -  ar*) 
for  the  required  integral. 

(6.)  If  M^a?  4-  Nc?y  =  0  is  a  homogeneous  function  of 
X  and  y  of  the  degree  n,  its  variables  a?  and  y  may  be  sepa- 
rated. 

For  if  we  divide  M  and  N  by  a?",  it  is  manifest  that  the 
equation  will  be  reduced  to  the  form 


/(l)'^-+/'(l)'^^  =  «' 


BETWEE^^   TWO   VARIABLES.  461 

since  it  is  clear  that  the  dimensions  of  y  in  the  numerators 
of  the  quotients  equal  those  of  a?  in  the  corresponding  de- 

nominators.     If  we  put  -  =  2  or  y  =  a?^,  we  have 

dy  =  zdx  +  xdz^ 

and  thence  our  equation  is  reduced  to 

f{z)  dx  +f  {£)  {zdx  +  xdz)  =  0, 

or  [f{z)  +  zf{z)-\dx^-xf{z)dz, 

dx  f'{2)dz 

or  its  equivalent     —  —  —     '  .  ,      /.,  tt? 
X  f{z)-\-zf{z) 

in  which  the  variables  are  separated ;  consequently,  we  shall 
have  log  x  =  —    I  - 


f{z)+zf'i^) 


EXAMPLES. 

1.  To  find  the  integral  of  (af^  +  yx)  dy  =  {xy  +  y^)  dx. 
Dividing  by  x^,  we  have 


,         ydx 
or  dy  =  - — ,     or 


,         ydx  dij       dx 

III  =  ^ — .     or     -^  —  — , 


y         X 

which  gives  log  y  =  log  cx^  <y£  y  ■=  cx\  which  results  im- 
mediately from  the  proposed  equation,  by  erasing  the  factor 
X  ■\-  y  that  is  common  to  its  members. 

2.  To  find  the  integral  of  [y  +  \/{q(?  —  y-)]  dx  =  xdy. 

Dividing  by  x^  we  have 


462  EQUATIONS   OF  TUE   FIRST   ORDER 

and  thence,  from  the  formula,  we  have 

]oga;=    /*— — 4:===  sin-^s  +  C  =  sin-^^  +  C. 

3.  To  find  the  integral  of  (^/x  -f  y-)  dx  =  (a?-  —  xy)  dy. 
Dividing  by  a?'^,  we  have 

{z  +  5-) dx  =  {i  —  2)  dy] 

and  thence,  since    2  +  z^  =f{^)      3,nd      1  —  2  =  —fi^)i 
we  shall  have,  by  the  formula, 

2\ogx  =  J  dzi-^—-),     or    loga;2+ log3  +  -  =  0, 

or  — f-  log  xy  =  C, 

as  required. 

4.  To  integrate  (  Vx^  +  y-  -\-  y)dx  =  xdy. 
Dividing  by  x,  we  have 


{Vl-i-2'-\-z)dx  =  dy. 
Hence,  by  the  formula,  we  shall  have 

log  X  =    f-—l=^  =  log  C{2+   |/?+l), 

or  x'^=c[y+  V{y'-^^')l 

which  may  evidently  be  changed  to  the  form 

{a?-cyr  =  G'(f+^\ 

or  its  equivalent  a?-  =  2Gy  +  c^. 

5.  To  integrate  {x  +  sy)  dx  +  yc/y  =  0. 

Here  (1  +  2,?)  cZ^  +  2dy  =  0, 

/(3)=r  1+  22,     and    /(5)=s; 

and  thence  by  the  formula  we  shall  have 


BETWEEIT  TWO   VARIABLES.  463 

•2/ 

or  we  have  log  (x -\-  y)  -^ =  0. 

(Y.)  Equations  between  x  and  y  maj  sometimes  be  made 
homogeneous  by  certain  substitutions,  and  thence  their  in- 
tegrals may  be  found.     Thus,  if  in 

{mx  +  ny  -\-  p)dx  -\-  {ax  -^  hy  ■{-  c)  dy  =z  0, 
we  put  x  =  x'  +  A,     y  ~  y'  -{-  B, 

and  assume 

Am-{-Bn+p  =  0,     Aa+Bh+e  =  0, 

we  shall  have  the  homogeneous  differential  equation 

(mx'  +  ny')  dx'  +  {ax'  -f  ly')  dy'  —  0, 

whose  integral  can  thence  be  found. 
Solving  the  equations 

Am  H-B7i-fi?=  0,     Aa  +  B5  4-c  =  0, 

T  .        en  —  hp         -y     -r,       (fP  ~  ^^<? 

we  have     A  —  —^ ~     and    ±5  ==  -^ , 

7)10  —  an  mo  —  a7i 

which  give  the  values  of  A  and  B  when  mb  —  an  is  different 

from  nauffht;  but  when  mh  —  a7i  =  0,  we  have  J  =  — , 
which  reduces  the  proposed  equation  to 

{rnx  +  ny  +^)  dx-\-  {ax  +  by  +  c)  dy  = 

{mx  +  ny  -{-_p)  dx  -\ Inx  -f  7iy  -\ )  dy  =  0, 

or  jpdx  +  cdy  +  {rnx  +  7iy)  .ldx-\-  —  dy)  =  0 ; 

which,  by  putting  7?ix  -\-  ny  —  z,  gives 


464:  EQUATIONS  OF  THE   FIRST  ORDER 

mdxc  +  ndy  =  dz    and    pdx  -f  cdy  +  2  idx  -\ dy\  =  0, 

or  we  have       (p  -{■  2)  dx  -] dy  =  0. 

Hence,  since    dy  =  — ,  we  readily  get 

dx  -f  (7/ig  +  a2)d2  ^  ^ 

mnp  —■  m-  c  +  ipii^  —  d^i)  2          ' 
in  wliicli  tlie  variables  are  separated;   and  if  71  =  «,  this 
reduces  to  the  very  simple  form 

,     ,    {viG  +  az)  d2        _ 
m7ip  —  vr  0 
(See  Lacroix,  p.  253.) 

(8.)  Particular  cases  of  integrability  of  differential  equa- 
tions between  x  and  y  may  often  be  discovered  by  reducing 
them  to  homogeneity. 

To  illustrate  this,  let  there  be  taken  the  equation 

dy  -\-hy^dx  =  ax"^  dx, 

called  the  equation  of  Blccati. 

1.  If  7n  =  0,  the  equation  is  equivalent  io  dx  =^ 


a-b/' 

in  which  the  variables  are  separated,  and  of  course  it  is  in- 
tegrable.     Indeed,  since 

we  easily  get 

a'  +  h'y       a'—h'y 
whose  integral  is 

2a^x  =  \  log  {a^  +  l>'y)  -  \  log  (a^  -  h^  y)  +  C. 
b'  b' 


BETWEEN  TWO   VARIABLES.  465 

2.  If  m  is  different  from  naught,  we  mav  put  y  =  z^^  and 
thence  get  dy  =  kz^~^d2  ;  consequently,  from  the  substitu- 
tion of  the  values  of  y  and  dy  in  the  proposed  equation,  we 
have  kzd2^~'^  +  h2"''dx=.ax'"\ix. 

To  make  this  a  homogeneous  equation,  we  must  equate 
the  exponents  of  z  and  a*,  and  we  shall  have  k  —  l=2^=w, 
or  ^  =  —  1  and  7?^  =  —  2  ;  consequently,  the  equation 

dy  -\-  h/dx  =  ax^dx 

becomes  integrable  when  we  put  z  ~  ^  for  y,  and  —  z  for  m, 
and  is  reduced  to 

—  z~-dz  +  'hz~^-dx:=^  ax~^dx^ 

.     T     ^           dz       hdx       adx 
or  its  equivalent ^  H s-  =  — s-« 

^  Z^  Z'  X- 

3.  If,  withLacroix,  at  p.  256  of  his  "Calcul  Integral,"  we 

put  y  =  '^_j  +  J- ,  we  shall  have 
x"       ox 

and  thence  we  shall  have 

or,  since  c??/  +  hy^^x  =  ax"'dx, 

we  shall  have  -~  +   ^  ^     =  ax^'-dx. 

x^  x^  ' 

or  c//  +  5/2-^  =  aa;'"+2^; 

Si/ 

which,  by  putting  a;  =  ~  becomes 
dy'  -  ly'Hx'  = 

20* 


466  EQUATIONS  OF  THE  FIRST  ORDER 

or  putting  —  y'  for  y\  we  shall  have 

dy'  +  hy"-d,Xi'  =  ax'-"'-^ 
which  is  an  equation  of  the  form 

dy  +  hy-dx  —  wx^dx  ; 

and  becomes  integrable,  as   before,  when   7/1  -f-  4  =  0,    or 
when  m  =  —  4,  and  is  obtained  immediately  from 
dy  +  hyHx  =  ax'^^dx, 

by  putting  y=-|,+  ^     or    y  = -y'x" +  j,  whenx' hi 

put  for  -  . 

It  is  hence  clear  that  the  equations 
dy  +  hifdx  —  ax-'''- ^dx     and     dy'  -f  h/"dx'  =  ax' -"'-  ^dx\ 
are  of  such  a  nature,  that  if  in  the  first  we  put 

a?  =  —     and     y  z=  —  y'x'^  +  -r  , 

it  will  be  changed  into  the  second ;  and  that  if  in  the  second 

1  x 

we  put  x'  =  --    and     y'  =z  —  yx^  +  ^ , 

X  0 

it  will  be  changed  into  the  first ;  consequently,  either  of  the 
equations  is  a  transformation  of  the  other. 

4.  Eesuming  the  equation 

dy  +  hy'^  dx  =  ax^  dx, 

and  putting  y  =  ±  — ,  we  have  dy  =  ^^  -^  ;   and  thence 
we  get 

T  ^  H ^  =  ax^dx,      or      T  dy'  +  Idx  =  ay'^x'^dx. 

If  we  put  a?"*  + 1  =  a?',  we  have 


BETWEEN  TWO   VARIABLES.  467 

— —  dx' 

X  —  x'  "'^K         x"'dx  —  — ^— r; 

and  thence  the  preceding  equation  is  easily  reduced  to 

7  —  m 

^    ^        7/1  +  1  m  +  1 

T  —  Wl 

or  to       dy'  ± .  y""  dx'  =  ± x'"'  +  ^  dx'. 

It  is  manifest  that  if  m  =  —  4  in  the  equation  of  Riccati, 
that  it  will  be  integrable,  and  thence 

7  — wi 

dy'  ±  -%  y"dx'  =  ±  — — ,  x'  "^  ^'  dx' 
^        m  +  l^  7?^+l 

derived  from  it,  and  having  the  same  form,  by  putting 

V  =  ±  -7     and     a? "'  ^  ^  =  x', 

must  also  be  integrable ;  that  is  to  say,  the  equation 

dy  +  hy'dx  =  ax~^dx 
being  integrable,  it  follows  that 

dy'  ±  3-3  y"dx  =  ±  -"3  r/''  ^a?' 

must  also  be  integrable,  and  thence,  by  putting 

—  m  —  4  =  —  -     or      7?2  =  g— 4=  —  g, 

is  the  value  of  m  for  another  integrable  case ;  and  putting 
—  -  for  7/1  in  the  equation 


T  y  -dx  =  ih , 

7/i  +  1  ^  m  +  1 


C^^'  ±    —   -^    7/^Va?^  =    ±    -— --r    ^^»*  +~1    dx', 


g 

we  have  ??i  ==  —  -  for  the  value  of  m  in  another  intearrable 
o  ° 


468  EQUATIONS  OF  THE   FIRST  ORDER. 

case  of  the  equation  of  Kiccati,  and  so  on ;  noticing,  that  the 
general  form  of  the  exponent  m,  when  0  and  —  2  are  not 

included,  is  m  =  —  - — --zr .  which  is  called  the  CHterion 

of  Integrahility  of  BiccaWs  Eiiuation^  q  being  any  number 
in  the  series  1,  2,  3,  4,  &c.  * 

It  may  be  noticed,  that  all  the  terms  that  result  from 
taking  —  for  ±  in  the  denominator  of  the  criterion,  must  Jdc 
considered  as  resulting  from  the  equation 

dy  -\-  hy^  dx  =  ax-"^  ~  '^  \ 

while  those  terms  that  result  from  taking  -|-  for  ±  in  the 
denominator  of  the  criterion,  must  be  supposed  to  have  re- 
sulted from  the  equation 

dy'  ±  -z  y'^^dx'  =  ± x'^  + 1 . 

5.  To  perceive  the  use  of  what  has  been  done,  take  the 
following 

EXAMPLES. 

1.  To  find  the  integral  of  dy  +  y'^dx  =  a^x-^dx. 
Here,  by  putting  g  =  1  m  the  criterion,  and  usiDg  —  for 
±  1  in  its  denominator,  it  becomes 

-4  . 

""'■  ^  2"=ri  =  -  ^' 

which  agrees  with  the  exponent  of  x  in  the  right  member 
of  the  proposed  equation,  and  of  course  shows  the  equation 
to  be  integrable.  To  perform  the  integration  we  proceed,  as 
at  p.  466,  by  putting 

X  —  ~     and    y  =  —  y'x'^  +  x\     since     5  =  1, 

X 

and  thence  get    dy'  -\-  y'Hx'  —  a^x'-'^dx^ 


BETWEEN   TWO   VARIABLES.  469 

as  at  p.  4G6 ;  where,  by  regarding  —  4,  the  exponent  of  x  in 
the  right  member  of  the  proposed  equation,  as  being  equal 
to  —  m  —  4  the  exponent  of  a?  in  the  equation 

dy  +  htfdx  —  a-x~"^~'^dx, 
we  shall  have  m  =  0;  and  thence  the  preceding  equation 
reduces  to  dy'  +  y^'^dx'  =  a^dx', 

which  gives 

&'  =  -^'-^^  =  (^,  +  JI-)  ^  2a. 
a^  —  y  ^        \a  -\-  y         a  —  y  I 

Integrating  this  equation,  we  have 

"lax'  =log  C  ^^^,     or    e^^^'  x  ^-^  ==  C  ^  const 

From  X'  =.  -  and  y'  =l  —  yx^  +  a?,  we  get 

X 


/  x(xy-l)-a   \  ^  ^ 
\a?  i—xy—  1)  —  a/ 


for  the  required  integral. 

2.  To  find  the  integral  of  dy  +  y'^dx—  —  a'x-Hx, 

Putting  a?  =  -  and  y  —  —  y'x'^  +  a?',  we  get,  as  in  the 

X 

preceding  question,  dy'  +  y'^'-dx'  =  —  c^dx' . 

cly^ 
Hence       dx'  = —  — ■ ^ 


a^  +  y'' 


'■Mfl 


.-1  y . 


whose  integral  gives    aa^^  +  C  =  cot'  , 

or,,  since  x'  =  -     and     y'  —  —  yy^-  +  a?,  we  have 

X 

X  a 


470  EQUATIONS  OF  THE  FIRST  ORDER 

3.  To  find  tlie  integral  of  dy  +  ifdx  =  2x~  *dx. 

Here,  by  putting  $'  =  1,  and  using  +  for  ±  in  the  de- 

4 
nominator  of  the  criterion,  we  have  m  =  —  - ,  and  of  course 

we  must  compare  the  proposed  equation  to  the  equation 

a                               h         —-- 
dy'  ± y^'^dx'  =  ± ^  x'"'  +  ^  dx\ 

given  on  p.  467 ;  consequeatly,  we  shall  have 

1, =  2,     and 


m  +  l~    '     m  -f  1  ""    '  7/1  +  1  ~  3  ' 

agreeably  to  what  is  said  at  p.  468  ;   hence, 

Srn  =  4:m  +4     or    m  =  —  4,     a  =  77i  +  1  =  —  S, 

h  =  2?n  +  2  =  —  6, 
and  thence  we  get 

dy"  -  ^"\     dx"  =  -  Zx"-'dx'\ 

Hence,  from  the  formulas  at  p.  466,  we  have 

dy'"  -^y"'Hx"'  =  -Ux'"  ', 

since  —  m  —  4,  the  exponent  of  a?  is  here  —  4,  and  of  course 
7n  =  0.     From  this  equation  we  have 

whose  integral  is 

1        y'"  +  A 

Because       x'"  =  ^  and  y'"  =  -  y"x"'  -  ^  , 
and  from  the  formulas  at  p.  466, 


BETWEEN   TWO   VARIABLES.  4:71 

1  _i_ 

2/  =  ±  -, ,        »■'"»  + 1  =  x'\ 
we  here  have 

x'"  =  4,    y"  =±  \ ,    and    x"  ^x'^K 

Hence,  since  ??^  +  1  =  —  3,  we  have 


-^    and    x"'z=.\^-l- 


or,  since  ij?  =  -, ,  we  have  a?   ^  =: ,  and  thence  x'"  =ix    ^ ; 

and  from 

y"'  =  -y'V'^-y,   2/"=^,,    and    y'  =  -ya==+c., 
we  have 

,,,  _  _       ^'"^       _  ^  —  _  ^  +  a?^(l  — ya?) 

Hence,  from  the  substitution  of  the  values  of  x'"  and  y'"  in 


1 


lo^C 


y-  + 


i/^ 


6i/2     °      .,,„      ./I' 
we  shall  get 


y"'-V'--. 


6  |/2  6  +  a;^  (3  4/2  +  o^'O  (1  -  yx) 


—  L 


or  6.^/2^)    '  =  log<?^'^^^   ^ 

6^x-W^  +  ^' (3  i/2  +  a?^)  (1  -  ya?)\ 
gives  e^^''   ^  I p^ — r^^-^ ^— ^1  =  const 

^6  -  x^  (3  V2  -  x-^)  (1  -  ya;)^ 
for  the  sought  integral. 

4.  To  integrate     dy  —  y\ix  =  2x~^dx. 
Comparing  the  equation  to 


472  EQUATIONS  OF  THE  FIRST  ORDER 


we  have 


a                       h  -"^ 

du'  ± T  y''dx  = «;'"»  +  1  dx, 


1,     -^T-2,     and        ™ 


///2\  > 


7/1  +  1  '      7/i  +  1  '  7;<.  +  1  ~  3 ' 

whicli  give  7/i  =  —  4,  a  =  3,  and  5  =  —  6,  and  thence  we 

have  dy"  -  Qy"'dx"  =  %x"-Hx". 

Hence,  from  the  formulas  at  p.  ^^^^  we  have 

dy"'  -  ^y""dx"'  =  Zdx'", 
which  gives 

whose  integral  gives 

3-v/2aj'''=  tan-V''V2  +  0. 
From 

a?'''  =  —r.    and    a;''  =  aj'^^^n  =  a.'~3  _  — 
x"  ^  ^-V 

we  have  x'"  =  x* ; 

also     ,"-_,'v4and,"  =  ^  =  -^, 

we  have 

4  JL  2. 

,„  _  X*  x^  __  6  x^  (yx  —  1) 

y     -  "  -y^  +  x        6   ""  e^^  (yaj  -  1)  ~  6a;^(y^~l) 

X*  —  ya?*  +  6 


Hence,  we  shall  have 


6a?  *  {yx  —  1) 


if-  =  tan-  ^^-r^^  +  ^    +  0 

aj'  3  \/2x'  {yx  —  1) 

for  the  required  integral. 


BETWEEN"   TWO   VAETABLES.  473 

(9.)  It  may  be  added,  that  differential  equations  may  often, 
by  the  introduction  of  new  variables  and  particular  processes, 
be  reduced  to  integrable  forms. 
1.  Thus,  to  find  the  integral  of 

jpdx       rdy  x^dx 

X  y    ~    «^"  ' 

since  the  integral  of  the  terms 

^  +  ~-    IS    log  xHf. 
X  y 


by  putting  x^y^  =  2  we  have  y"-  =  —  ov  y  =  ^— j  , 

n 

thence  2/"  =  (-^1    ;  consequently,  the  proposed  equati( 


1 
and 


mr  +  7ip 

reduced  to      d  log  x^y''  =  d  log  z  = , 

n 
Z^dz  ?»>•  +  riTp  Vl  —  \  >nr  +  np 

or =  a?     ""      dx,     ov    z''     dz  =  x     '"     dx, 

z  ' 

in  which  the  variables  are  separated.     Integrating  this,  we 
have 


const., 


n 
Zr 

mr  +  np  +  r 
X          r 

n 

7rir  +  np  -\-  r 

T 

n 

r 

mr  -^  np  -y  r 

IZr 

X           r 

or  = h  const 

n  mr  -\-  np  -\-  r 

Kestoring  the  value  of  z^  we  have 

m.r  +  np  +  2 


■np 


nx 


au'x  ^  = , 

^  mr  -\-  np  +  r 

which  needs  no  correction,  supposing  y  and  x  to  commence 


474  EQUATIONS  OF  THE   FIRST   ORDER 

np 

together  ;  dividing  the  members  of  this  equation  by  ic  »• ,  it 
is  immediately  reduced  to 


aif 


nx 


m  +  1 


7nr  +  np  +  r 

Remarks. — The  preceding  method  of  finding  the  integral 
is  analogous  to  that  of  Lacroix,  at  p.  259  of  his  "  Calcul 
Integral." 

The  integral  can  also  be  immediately  found  by  mulliplyii  .g 
its  members  by  -  a?  »•  y'\  which  gives 

—  y^x''      dx-\-7ix  "- y"'-^  dy  =  d  ix  *•  y"^]  =  —  x'    '•"    dx; 
whose  integral,  as  above,  is 


np 


mr  +  np  +  r  m  -f- 1 

nx       ^  nx    »■ 


ax  ^  y"= or     ay"  = , 

mr  -\-  np  -\r  T  'rnr  -\-  njp  ■\- r 

supposing  the  integral  to  commence  with  x. 
2.  To  integrate  the  equation 

dy       dx  _  x^dx 
y         X        ay  \/n ' 

we  multiply  its  members  by  -  ,  and  thence  get 

dy       ydx  _  x"^~'^dx  . 
X  x^    ^     a  |//i    ' 

an  exact  differential.     Taking  the  integral,  we  have 

y  a^  g^m  +  i       . 

-  = —     or     y  =  — ; 

X       ma  \/n  ^        ma  \/n 

which  needs  no  correction,  supposing  the  integral  to  com- 
mence with  X. 


BETWEEN  TWO   VARIABLES.  475 

8.  To  integrate  tlie  equation 

,  .  dx  dy 

we  may  clearly,  from  what  is  shown  at  pp.  3-i  and  35,  take  the 
differential  of  its  members  by  regarding  dx  as  being  constant, 
and  shall  thence  get 

,         d-ydx  .  N        7     ,    7         1        d'^y 
d^  ^"^  ^'  y)  =  dx  +  dy~dy-  -^  X] 

or,  by  reduction,  we  shall  have 

dx  ,  ,         X  dx-  dip- 

dif^        c/;        ^y^j  ^         a -\- y' 

in  which  the  variables  are  separated.     Hence,  we  shall  have 

dy       dx 

{a  +  yY  X' 

whose  integral  gives 

(a  +  2/)^  =  ±  a?^  +  c  ; 
or,  by  squaring, 

y  -{-  a  —  x  ±_  2cx^  +  c^, 
which  can  be  further  reduced  to 

iy  +  a  —  x^  —  a-f  zsz  4:o\ 

which  represents  the  integral  of  the  proposed  equation,  taken 
in  its  most  general  sense. 

4.  To  find  the  integral  of  ady  —  ydx  —  xdx. 

By  assuming  y  ==  a  4-  v  +  a?,  we  have  dy  =:  do  +  dx^ 
and  thence  by  substitution  the  equation  becomes 

adv  -f  adx  =  adx  -f-  vdx  +  xdx  —  xdx  ; 
or,  by  erasmg  the  terms  that  destroy  each  other,  we  have 

—  =z  dx   whose  intep^ral  is  a?  =  a  looj  cv ;  or,  since 


476  EQUATIONS   OF   THE   FIRST  ORDER 

V  ^=  y  —  a  —  x^ 

-we  sliall  have  x  =  a  log  c  (y  —  a  —  x).  (See  Vince's 
"Fluxions,"  p.  181.) 

(10.)  We  will  now  show  that  if  we  have  a  differential 
equation  of  ^idx  +  ^dy  =  0,  of  the  first  order,  between 

two  variables  x  and  ?/,  in  which  the  condition  -r~  =  — r-  of 
^  dy         dx 

integrability  is  not  satisfied,  tliat  the  condition  may  still  be 
satisfied  after  it  has  been  multiplied  by  a  suitable  factor; 
and  of  course  the  integral  can  be  found. 

For  since  ^dx  +  ^dy  =  0  is  not  considered  as  being  im- 
mediately integrable,  it  may  be  supposed  to  have  been 
obtained  by  eliminating  a  constant  from  an  equation  of  the 
form  F  (a?,  y)  —  0  and  its  first  differential.  Hence,  if  C 
stands  for  the  constant,  by  solving  the  equation  with  reference 
to  C,  we  shall  obtain  an  equation  of  the  form  C  ^^f{x^  y) ; 
consequently,  by  taking  the  differential  of  this,  we  shall, 
without  reduction,  get  the  differential  equation 
M'c^^  +  Wdy  =  0, 

in  which  -/  or  -y-  must  clearly  be  the  same  as  in 
dx       ay  ^ 

Mdx  +  ^dy  =  0, 
since  the  two  equations  result  from  the  elimination  of  the 
constant  C,  from  the  equation  F  (x,  y)  =  0  in  two  different 
ways ;  the  proposed  equation  resulting  from  the  elimination 
of  C  from  F  {x,  y)  =  0  hj  means  of  its  differential  equation, 
and  the  equation  M'dx  +  N'c?y  =  0  resulting  from  the  im- 
mediate differentiation  of  the  equation  G  =zf  (ce,  y). 

Hence,  eliminating  ~  from  the  preceding  equations,  we 

V  n      ^  dy  M  -     dy  W 

Bhallget         £  =  -^    and    -£  =  -^^ 


BETWEEN   TWO    VARIABLES.  477 

M      M^ 

consequently,  we  get  ^  =  — -,  such,  tliat  M^  and  N'  must 

clearly  be  like  multiples  of  M  and  N. 

Eemarks. — 1.  Having  found  M'  and  N',  it  is  manifest  that 
the  integral  of  Wdx  +  Wdy  —  0  will  give  C  =/(«,  y\  in 
which  C  represents  the  arbitrary  constant,  and  which  rep- 
resents nearly  a  transformation  of  the  equation  F  (a?,  y)  =  0. 

2.  Since  M'dx  +  ^^dy  =  0  is  an  exact  differential,  it 
follows,  from  Euler's  Criterion  of  Integrability  (see  p.  440), 

,    ,,  ,  ^M'       dN' 

that  we  shall  have  —7-  =  -7-. 

ay         ax 

Hence,  if  z  represents  the  factor  of  M  and  N,  which  gives 
Ms  =  M'    and    N.s  =  N',    the   condition  of  integrability 

dUi     dm 
becomes  -^-  =  —j-  , 

ay  ax 

which  gives 

(Mr/s  +  zdM)  -^dy  —  (Nc^s  +  ^dl^)  -v-  dx, 

dM.       (iN\  __-^  d3       ^dz 
dy        ~dx  I  dx  dy"* 

which  z  must  satisfy.  Having  found  M.zdx  +  '^zdy  =  dic^ 
it  is  manifest  that  the  members  of  this  multiplied  by  any 
function  of  u  will  also  be  an  exact  differential ;  consequently, 
there  will  he  an  unlimited  nmnber  of  factors  that  will  make 
the  proposed  differential  an  exact  differential. 

EXAMPLES. 

1.  To  find  the  factor  which  will  reduce  ydx  —  xdy  =  0 
to  an  exact  differential. 

Here  we  have     M.  =:  y    and    K  =  —  a?,     and  thence 


or  z  ( ^  —  -~  1  =  N  ^  —  M 


^  _  ^\  -  N  ^  -  M  — 
dy        dx)  ~       dx  dy"* 


478  EQUATIONS  OF  THE   FIRST   ORDER 

which   can  clearly   be   satisfied   by   putting    2  =  -3,    and 
gives  -^  =  ^, ,  an  identical  equation ;  or,  by  writing  the  form 

if  J 

^  =  0,  and  intesratinff 

X         y  ' 

ydx  —  xdy        _  .in  dx       dy 

^ ,- — ^  =z  0,     or  the  form -, 

y^  a?         y  ' 

X 

we  have  -  =  C  =  const. 

y 

X  X  £C  X 

Kemarks. —     -  (^  -  =  0 ,    —  cZ  -  r=  0,    and,  generally, 

y    y         r    y 


i> 


V/     y 


are  also  exact  differentials  of  the  proposed  equation,  agree- 
ably to  what  has  been  done. 

2.  To  find  the  factor  that  reduces  ydx  —  mxdy  =  0  to 

an  integrable  form. 

Here,  as  in  the  preceding  example,  we  get 

1                ,   1              ydx  —  mxdu        ,  x 
^  =  -^-n  1     and  thence zr—r-^  =  d  —- 

ytn  +  1  7  ytn  + 1  ytn 

is  the  transformed  differential,  whose  integral  is 
-—  =  C  =  const. 

8.  To  find  the  factor  that  makes  dy  +  Vydx  =  Qdx  inte- 
grable. 

Here  M  =  Py  —  Q    and     N  =  1,  and  thence  we  have 

-J 7-  =  P     and    2P  ==  N  -7-  =  -T-  » 

dy         clx  ax       ax 

supposing  2  to  be  independent  of  y ;  consequently,  we  have 

dz  /* 

—  =  "PdXj  whose  integral  is  log  2  :=   I  Fdx,   supposing    the 

constant  to  be  included  under  the  sign  of  integration    /  . 


BETWEEN"  TWO   VARIABLES.  479 

Multiplying  the  proposed  equation  by   s  =  ^/  ^"^^^  which 
gives  log  B  =J  ^'^^  we  have 

whose  integral  is  eJ  ^^""y  =   I  cJ^'^^Q/ix^ 

and  thence  y  =  e-f^'^4  CeA'^^Qdxj, 

supposing  the  constant  to  be  indicated  by  the  preceding  sign 
of  integration,  or  the  integral  may  be  expressed,  as  at  p.  456, 

by  y  =  e  '/^"^(^feA'^Qdx  +  C'\  , 

4.  To  find  the  factor  that  makes 

aryy  +  l4:aPy ==^J  ^^  =  0 

integrable. 

Here      M  =z  4a?-y -:, ^,     and     N  =:  a?^, 

and  thence  we  shall  have 

ay        ax 
consequently,  supposing  s  to  be  a  function  of  x  only,  we 

shall  have  zx^  =  x^ -p^    or     ~  =  — , 

ax  2  X 

which  is  clearly  satisfied  by  putting  2  =  x.     Hence,  multi- 
plying the  proposed  equation  by  a?,  we  have 

whose  integral  is  a?'*y  +  |/  (1  —  x^)  —  C. 
6.  To  find  the  factor  that  will  make 

aydy  +  (px  —  hy-)  dx=^  0 
an  exact  difierential. 


480  EQUATIONS  OF  THE  FIRST  ORDER 

Ilere  'M.  =  ex  —  h/    and    N  =  ay, 

and  thence  —^ ^  =  —  2hy. 

dy        dx  ^' 

dz 
wtiicli  gives  —  s  x  2hy  =  ay  ^  ^ 

by  supposing  z  to  be  independent  of  y ;  which  gives 

dz  2hdx  -=^" 

z  a 

for  the  sought  factor.  Hence  the  transformed  differential 
becomes 

26a; 

\_aydy  +  {ex  —  hy-)  dx']  e~  "    =  0  ; 
whose  integral,  sometimes  called  the  prir/iitive,  is 

(See  Young,  p.  210,  &c.) 

(1 1.)  We  now  propose  to  show  how  to  integrate  any  homo- 
geneous differential  equation  consisting  of  any  number  of 
variables. 

Thus,  let    Udx  +  Nf7y  -f  Fdz  +  &c.  =  0 

be  a  homogeneous  differential  equation,  consisting  of  any 
number  of  variables ;  then,  if  the  equation  is  not  integi-a- 
ble,  it  is  clear  from  what  is  shown  at  p.  445,  that  it  must  be 
on  account  of  the  omission  of  a  homogeneous  factor,  com- 
mon to  its  terms.  Hence,  if  ic  stands  for  the  omitted  factor, 
we  shall  have 

uMdx  +  uNdy  -f  uTdz  -\-  ko.  =  du'  =  0, 

the  differential  being  exact.  If  n  denotes  the  degree  of 
homogeneity  of  u\  we  have,  from  what  is  shown  at  pp.  445 

and  446,       wM^  -f  i/Nt/  +  wP^  +  &c.  =  qui'] 


BETWEEN  TWO   VARIABLES.  481 

consequently,  dividing  the  members  of 

yMdx  +  wN%  +  &c.  =  du' 

by  the  members  of  the  preceding  equation,  we  shall  have 

yLdx  +  l^dy  +  &c.  _  du'  ^ 
M.X  +  Ny  +  &c.  nu' ' 

consequently,  since  the  right  member  of  this  is  an  exact 
differential  (its  integral  being  -  log  w'),  it  is  plain  that 

l^dx  +  l^dy  +  &c. 
Ma?  +  Ny  +  koT 

must  also  be  an  exact  differential. 

It  hence  follows,  that  the  factor  which  makes  the  proposed, 
differential 

Mc?a?  +  Nc^v  +  &c.  =  0  exact,  is    ^^^ -^r-f — ; — -. — ; 

^  '         Ma?  +  JN  y  -f  &c. 

and  thence^  if  lA.dx  +  l^dy  =  0  is  the  proposed  equation^ 
tlie  requisite  factor  is  y^ :^. 

Remarks. — 1.  It  is  clear,  from  pp.  445  and  446,  that  the 
degree  of  homogeneity  of  Ma?  +  Ny  +,  &c.,  when  the  pre- 
ceding process  is  applicable,  must  be  different  from  naught ; 
and  Mii?  +  Ny  +,  &;c.,  must  also  be  different  fi'om  naught, 

2.  If  Isidx  +  Nc?y  =  0,  and,  at  the  same  time,  Ma?+Ny=:0, 
then,  eliminating  N  from  the  first  of  these  by  means  of  the 
second,  we  shall  have 

which  shows  that  if  My  is  a  function  of  - ,  the  integial  can 

if 

be  immediately  found  in  its  most  general  form. 

21 


482  EQUATIONS  OF  THE  FIRST  ORDER 

EXAMPLES. 

1.  To  find  the  factor  that  makes 

{xy  —  i/^dx  -\-  {yx  +  ^)  dy  =  0, 

an  exact  dififerential. 

Since  Ma;  +  Ny  =.2d^y^ 

bj  dividing  the  given  equation  by  ar*  y,  we  have 
whose  integral  is         log  xy  ■\-  -  =^  Q,. 

X 

2.  To  integrate   {^  —  y'^)  dx  -\-  {xy  +  oF)  dy  =  0. 
Here  Ma?  +  Ny  =  aP  {x  -{-  y), 

and  thence,  dividing  the  given  equation  by  this,  we  have 

\x         X^/  X 

whose  integral  is  log  x  +  ~  =  G. 

X 

3.  To  integrate       ydx  —  xdy  =  0. 

Here  M  =  y  and  N  =  —  a;,  and  thence  Ma;  +  Ny  =  0 ; 
consequently,  from  what  is  shown  above,  we  shall  have 

Myd-  =  0,      or     fd-  =  0; 

"  if 

1        /  7'\  (  ir\       X 

and  this  multiplied  by    -^  <p  1-1     becomes     01-)  c?-  =  0, 

an  integrable  form,  since  0  (-)  represents  a  function  of  -. 


BETWEEN  TWO   VAEIABLES.  483 

4.  To  integrate  {x^y  —  y^x)  dx  +  yx'^dy  =  0. 

From     Ma?  +  Ny  =  ar'y,     we  have     ( -A  dx  -\-  —] 

whose  integral  is         log  «  +  ■-  =  C, 

X 

the  same  as  in  example  2. 

5.  To  integrate  {x'^y  +  y^)  dx  —  {x^  +  a??/^)  dy  =  0. 
Here    M  =  y  (a?^  +  y^-)    and     N  =  —  a?  (a?^  +  y^), 

and  thence  we  have     Ma?  +  Ny  =  0 ; 

consequently,  from  what  has  been  shown,  we  shall  have 

and  it  is  easy  to  perceive  that  -j  0  (-)  is  the  factor,  which 
makes  this  integrable,  since  it  reduces  it  to 
fx^       ^  \      /x\    ,  X        ^  (x\    ,  a? 

which  is  clearly  an  integrable  form,  since  F  [-1  is  supposed 

a?  /x\ 

to  be  a  function  of  - ,  at  the  same  time  that  0  ( - 1  also  de- 
y  V 

X 

notes  a  function  of  - . 

y 


SECTIOlSr  YIL 

INTEGRATION"  OF  DIFFERENTIAL  EQUATIONS  OP  THE  FIRST 
ORDER  AND  HIGHER  DEGREES,  AND  THE  SINGULAR 
SOLUTIONS  OF  DIFFERENTIAL  EQUATIONS,  ETC.,  BETWEEN 
TWO   VARIABLES. 

(1.)  It  is  sometimes  said  by  authors,  tliat  differential 
equations  of  the  first  order  and  higher  degrees  can  not  result 
from  the  immediate  differentiation  of  any  integral,  but  must 
arise  from  the  elimination  of  an  integral  power  of  a  con- 
stant from  the  integral,  by  means  of  its  differential  equa- 
tion. (See  p.  311.)  That  what  is  here  affirmed  is  not  uni- 
versally true,  may  be  proved  from  the  simplest  considera- 
tions. For  (see  p.  191)  in  finding  multiple  points  of  the  first 
kind^  we  differentiate  the  equation  of  the  curve  by  regarding 
the  co-ordinates  at  the  points  of  intersection  as  being  inde- 
pendent variables.  Thus,  in  finding  the  multiple  points  of 
the  curve  whose  equation  (see  p.  191),  is 

ai/^  +  cxy  —  hx^  =  0, 

by  proceeding,  as  directed,  we  have  found  the  differential 

equation  2ad(/^  4-  2cdxdy  —  Qhxdjr  =  0  ; 

which,  divided  by  2da^,  and  representing  -~  by^,  becomes 

dt/       G  dy       Shx         „       g  Shx       ^ 

dor       a  ax         a        ^        a-^  a 

in  which  —■  or  p  is  taken  on  the  supposition  that,  after  the 


INTEGRATION  OF   DIFFERENTIAL   EQUATIONS.  485 

differentiation,  dy  is  a  function  of  dx^  or  contains  it.  To  find 
the  integral  of  tlie  preceding  equation,  we  must,  by  a  re- 
verse process,  reduce  it  back  to 

lady"  +  Icdxdy  —  %xd:x?  =  0, 

whose  first  integral  is 

2aydy  +  cxdy  +  eydx  —  Shx'dx  =  0 ; 

and  then  the  integral  of  this  is 

ay^  -f  cxy  —  hx^  =  0, 

the  proposed  equation,  as  it  clearly  ought  to  be.  Solving 
the  equation 

or,  since  p  —  -^^  we  have 

1  .  ^        T  .                 cx       (c^  -\-  12abx\^ 
whose  integral  is  y  =  —  —  ±  ^ ^r-r-^ — ^  +  const. 

Act'  ovCfO 

Hence  L  +  ^V  -  (^l±i?«^)_' 

Mence  (^y  +  ^^j  -      ^q^^      , 

by  omitting  the  constant,  or 

2  ca^      cV  _  c«  +  36  6'%Z>,T+  86^  X  12-a-Z'^,^'2  +  12VJV 


a 


4fr  3^  X  12'a'b 


which  clearly  can  not  be  reduced  to  the  integral 
ay^  +  cxy  —  hx^  —  0, 

or  the  proposed  equation.  If  we  integrate  the  equation 
dy'^  —  a-dx^  =  0,  supposing  x  and  y  to  commence  together, 
by  either  of  the  preceding  methods,  they  will  be  found  to 
give  y^  =  aV ;  while  dy"^  —  axdx^  =  0,  integrated  by  the  first 


486         INTEGRATION  OF  DIFFERENTIAL  EQUATIONS 
method,  gives  y-  =  — - ,  and  integrated  by  tlie  second  method, 

o 

4  . 

gives  y'  =  Q  aar*,  which  does  not  agree  with  the  preceding 

integral 

(2.)  The  common  method  of  finding  the  integrals  of  equa- 
tions of  the  form 

dy""  +  Vdy^'-'dx  +  Qdy^'-^da^  +    .....+  Uc^"  =  0, 
or  its  equivalent 

consists  in  solving  it  like  an  equation  of  the  nth  degree,  by- 
regarding  --  as  the  unknown  quantity,  and  of  course  there 
will  result  ?i  equations  of  the  forms 

dx     ^      ^'     dx     ^  -^'     dx     -^    -^' 

and  so  on,  to  n  equations ;  _p,  j?',  p^\  &c.,  being  the  roots  of 
the  equation. 

From  these  equations  we  get 

y—  J  V^^  =  0,     y  —  J  jp'dx  =0,     y  —  J  2y"dx  =  0, 

and  so  on.     Hence  we  shall  have 

\y  -  Jpdx)  X  (y  -  fp'dxj  X  [y-  fp"dyj  =  0, 

which  may  be  taken  to  represent  the  integral  of  the  pro- 
posed equation ;  noticing,  that  each  of  the  factors  may  be 
supposed  to  be  corrected  by  the  addition  of  the  same  con- 
stant. 

For  the  method  of  integration  here  proposed,  the  reader 


OF  THE   FIRST   ORDER  AND   HIGHER  DEGREES.        487 

is  referred  to  Lacroix,  "  Calcul  Integral,"  p.  279,  &c. ;  Young, 
"Integral  Calculus,"  p.  224  ;  and  Lardner,  p.  318. 

EXAMPLES. 

1.  To  find  tlie  integral  of  2/  ^-^  +  2;:e  ^  -y  =  0. 

Eeducing  tlie  equation  to  the  form 

ydif"  +  2xdydx  —  ydx^  =  0, 

and  taking  tlie  integral,  regarding  x  and  y  as  independent 
variables,  we  liave 

1/  {7,7/  of  '7' 

^—^  +  2yxdx-\-iK?dy  —  ydxx=0     and    |-  +  ^y—y^  =0, 

found  on  tlie  supposition  that  x  and  y  commence  together, 
and  that  y  in  the  last  term  is  constant ;  but,  since  y  is  not 
constant  in  the  last  term,  it  is  clear  that  the  equation  has 
not  been  obtained  on  the  supposition  of  x  and  y  being  inde- 
pendent variables  ;  noticing,  if  the  last  term  of  the  equation 
had  been  x  or  any  function  of  it,  the  proposed  equation 
might  have  been  obtained  on  the  supposition  of  x  and  y 
being  independent  variables,  and  of  course  a  doubt  as  to 
the  true  origin  of  the  proposed  equation  would  have  been 
the  result. 

Hence,  solving  the  equation  on  the  supposition  that  x  and 
y  are  not  both  independent  variables,  we  have 


dy  _—x  -\-  Vy'^  -\-  x"^         ,     dy  __  —x  —  \/  {if  -f-  x^) 
dx  y  dx~  y  ^ 

which  may  be  put  under  the  forms 

and  by  taking  the  product  of  these  factors,  we  have 


488  INTEGRATION  OF  DIFFERENTUL   EQUATIONS 

whose  integral  is 


±  iY  +  ic"  =  a?  +  C,     or    y""  =:  2cx  -\-  CK 

2.  To  find  the  integral  of  dy^  ±  da^  =  ^dxdy. 
Integrating  on  the  supposition  of  the  independence  of  x 

and  y,  we  have 

^^  =xy  +  C,     or    f'±x'=  2xy  +  C. 

Remarks. — If  we  take  +  for  ±  in  the  proposed  equa- 
tion, we  have 

dy^  +  da^  =  2dxdy,     or     dy'^  —  2dydx  -\-  da^  =  0, 
or  dy  —  dx  =  0, 

whose  integral  is  j/  —  a?  =  C ;  the  same  as  by  the  preceding 
method.     If  the  proposed  differential  is 

dy^  —  dix?  =  2dxdy^ 

it  is^clear  that  the  integral  found  on  the  principles  of  the 
independence  of  x  and  y,  and  their  dependence,  as  in  the 
common  method  of  integration,  will  not  agree ;  consequently, 
the  origin  of  the  proposed  differential  is  doubtful. 

3.  To  integrate  a?  -^  +  a?  —  1  =  0,     or     -,—„  = 1. 

°  dx^  '  dx^       X 

Multiplying  by  da?^  and  integrating  on  the  supposition  that 
X  and  y  are  independent  variables,  we  have 

da? 
dy^  =  ~ dx^^     and  thence    ydy  =  dx  log  x  —  xdx, 

X 

which  integrated  again,  gives 

y^  x^ 

^  =  X  log  X  —  X  —  —  +  const, 


OF  THE   FIRST   ORDER   AND   HIGHER  DEGREES.        489 

or  if-  —  2x  log  X  —  2x  —  X'  -\-  C  ] 

consequently,  the  origin  of  the  differential  is  doubtful. 
Eemarks. — Mr.  Young,  at  p.  226  of  his  work,  finds 


y  =:  \^x  —  x^  —  tan-^y — 


X 

for  the  integral ;  a  result  very  different  from  the  preceding. 

(3.)  When  only  one  of  the  variables  x  ov  y  enters  the  pro- 
posed equation,  and  the  value  of  the  variable  in  a  function 

dy 
oi  -~  =  jp  can  be  found ;  or  if  ^  can  be  found  in  a  function 

of  the  variable ;  then,  in  solving  the  equation  in  the  com- 
mon way,  the  other  variable  can  be  found.     Thus,  having 

found  aj  =  F(»,     or    y=f{p\ 

^       ,         (!¥{]))  ,  ,      .         df{p)  , 

we  get      ax  —  — r^^  dp,     and    dy  =  ~j       ^P  j 

consequently,  from 

dy  =z  pdx,     or    dx  =  —, 
we  shall  get 

y  =  fpdx  =  fp  '^  dp,  OV   x=  /J  =  /^|>  dp, 

whose  integrals  will  determine  the  value  of  y  ov  x. 
For  integrating 

by  parts,  we  shall  have 

y=p'F{p)-  fF{p)dp; 

so  that  if  F  (p)  =    ^         ,  we  shall  get 

=    2   .    -t  —    /  T---^  =  -ir-—^  —  tan-^  p  : 
p^  +  1        J  1  +y       y  +  1  -^  ' 


y 

21* 


490         INTEGRATION  OF  DIFFERENTIAL   EQUATIONS 

1  /f^~i 

"which,  from  x  =  -5 ,  gives  «  =  y  ,  and  thence 

J9r+  1     ^         ^  X 


y  =  Vx-x''-  tan-^  |/1— -  +  G 


X 


the  result  quoted  from  Mr.  Young  in  the  preceding  example. 

If  the  equation  involves  such  high  powers  of  x  or  y  that 

it  can  not  readily  be  solved,  we  make  such  a  substitution  for 


du  dx       1       ^ 

-J-  =J7,     or    --  =  -  =p 
ax      -^  dy      p 


as  will  reduce  the  degree  of  the  equation,  so  that  x  ox  y 
may  be  found  by  the  common  methods  of  solving  equations. 

Thus,  to  integrate 

we  put  J  =  ,T^, 

and  thence,  by  substitution,  get 

a^  +  x^z  -\-  a?'2^  —  0,     or     x=  —  :j 5 , 

which  gives 

_  _      dz  Bz'ds     _  _  (1  -  22')  dz 

'"^  ~     1  +  2^  "^  (1  +  s^j'  ~     "(i  -  ^J  ' 

From  -^  1=  xz, 

ccx  * 

by  substituting  the  value  of  x,  we  have 

dy  z^  ,  ^       1 

and,  substituting  the  value  of  dx,  thence 

_  __  (1-  22')z'dz  __  __     Sz'dz  2zHz 

^  ~         (1  -f  zj     ~"     (1  +  ^y  "^  (1  +  ^T 


OF  THE   FIKST   ORDER  AND   HIGHER  DEGREES.        491 

whose  integral  gives 

1 1  p 

^  ~  2  (1  +  ^J       3  (1  +  s^  "^ 

Solving  this   equation  bj  quadratics,  regarding —^  as 

being  the  unknown  quantity,   we  shall  get ^3    in   a 

function  of  y,  whose  reciprocal  gives  1  -f  2^  in  a  function 
of  y,  which,  diminished  by  1,  gives  s^,  whose  cube  root  gives 
the  value  of  z.  Hence,  x  is  easily  found  in  a  function  of  y, 
as  required ;  noticing,  that  we  may  clearly  proceed  in  like 
manner  in  all  analogous  cases. 

(4.)  If  the  equation  involves  both  variables,  in  such  a 
way  as  to  make  its  terms  homogeneous  relatively  to  the 
variables,  then,  putting  ?/  =  a?s  in  the  equation,  if  ^i  denotes 
the  degree  of  homogeneity  of  the  equation,  its  terms  will 
be  divisible  by  a?** ;  and  we  shall  have  an  equation  in  terms 
of  z  and  p,  whose  highest  power  in  s  will  be  z"^. 

Hence,  if  the  equation  can  be  solved  with  reference  to  z, 
we  shall  have  z=:F  (p) ;  or,  if  the  equation  can  be  solved 
relativ:ely  toj?,  we  shall  get  p  =f{z).     Since  y  =  xz,  we 

have  dy  =  xdz  +  zdx  =  xdF  {p>)  +  F  {p)  dx^ 

dy  dF(p)        ^,    . 

,  .  ,      .  dx  d¥  (p) 

which  gives  — 


X    p-F{py 

whose  integral  can  be  found  in  a  function  of  p ;  and  thence 

from  y  =  xz  =  x¥  {p), 

by  eliminating  p^  we  get  y  in  a  function  of  x^  as  required. 


492         INTEGRATION  OF   DIFFERENTIAL  EQUATIONS 

In  mucli  the  same  way,  from 

p  =z  -^=f{2\     and  from     dy  =  xdz  -f-  zdx^ 

xdz    ,  J,,  .  xdz         „. 

we  have    -^-  +  z  =/(4     or     -^-  =f{z)  -  z, 


which  gives 


a 

dx  dz 


which,  integrated,  expresses  a?  in  a  function  of  2;  and  thence, 
from  y  =  xz,  we  express  y  in  a  function  of  z ;  consequently, 
eliminating  z  from  the  values  of  x  and  y,  we  shall  get  y  in 
a  function  of  x. 

EXAMPLES. 


1.  Given  y  —  xj)  =  x  Vl  +  j^Ho  find  the  integral 
By  putting  y  =  xz,  the  equation  reduces  to 

z  - p  -  Virrp,     or    z^p  +  Vl  +  pS 
which  gives  dz  =:  dp  -\ ^  ^      : 

1/1 +y 

and  from  dy  =  pdx  =  xdz  +  zdx^ 

T  <^aj  c?3  dp  pdp 

we  have      —  = = — ^ — ^-^^ , 

a?       p-z  Vl-hp^       l+i> 

whose  integral  clearly  gives 

log ;» =  -  log  {p  +  vrry)  -  log  i/rrp  +  log  o,- 

and  thence  we  have 

G 


X  = 


Vl+p'{p+  Vl+p')' 


and  since  z  =  p  +  Vj^+p\ 


we  have  xz  =  y  = 


G 


Vl  +y 


OF  THE   FIEST  ORDER   AND   HIGHER   DEGREES.         493 

consequently,  we  hence  get 


X  —  ^ 


c  +  Vc-  —  2/^ 
for  one  form  of  tlie  proposed  integral. 

Otherwise. — By  squaring  tlie  members  of  tlie  proposed 
equation,  we  liave 

y^  —  ^xyp  +  x-^^  =z  a^  -\-  x"p^ ; 
or,  erasing  the  terms  that  destroy  each  other,  we  have 
r  -  x" 


V  \ 


or,  since  y  =  xz^  we  shall  have 
z^  -\ 

,   ,              c?a?             dz  ,  ,        dx  2zdz 

and  thence     —  =  -ttt- reduces  to     —  = 


/(«); 


^       /C^)  ~  ^  ^  1  -{•  z^ 

whose  integral  is 

log  ^  =  log  ^-^,    or    <^  =  j^-., 


1/        .       , 

which,  since  0  =  - ,  is  immediately  reducible  to 

X 

X  {x^  4-  7/-)  :=z  x^  (?,     or    x^  +  2/"^  ==  c'x, 

Eemarks. — It  is  easy  to  show  that  this  integral  agrees 
with  the  integral  found  by  the  first  method,  since  it  can  be 
put  under  the  form 

G  +  Vg"  -  f  =  ^\     or     VT^f  =  -  -  c, 
a?  ^         X 

which,  by  squaring  and  an  obvious  reduction,  becomes 
a;2  +  2/"  =  2gx^  which  agrees  with  a?"  +  ?/«  z=  g'x^  when  we 
put  g'  for  2g.  For  the  first  of  these  solutions,  the  reader  is 
referred  to  p.  229  of  Mr.  Young's  work. 


IIM-^)') 


494         INTEGRATION"  OF  DIFFERENTIAL  EQUATIONS 

2.  To  find  the  integral  of  y^  —  ji?ar  =  jpy^. 

Putting  xz  for  y,  the  equation  immediately  reduces  to 

z^ 
z^  —p  =pz^^     wliicli  gives    p  =  T'Jr^  —f(^)' 

L  -\-  Z 

^  ,  dx  dz  dz  (1  +  2^) 

Hence,  we  have    —  =  ttt- =  —  -^ — r— ^ . 

X       f{z)  —  z  z^  —  z-  -{■  z 

Smce 5 5 = 5 -,     we  snail  have 

z^  —  z^  +  z  z       z^  —  z  ■\-  V 

-dz 
dx  _       dz       dz  _  _  ^  ^ 

aj"~        z        z^  —  z  +  1  ~        z 

wliose  integral  gives 

,  A  -  4A 

log  K  =  —  log  Z  —  g^-j  =  —  log  C3 g-  , 

in  which  A  is  an  arc  of  a  circle,  whose  radius  =  ~-~  and 

1  1/  y 

tangent    =  z  —  -.     Since   s  =  -,    if  we  put  -   for  z  m 

^  XX 

this  equation,  we  shall  have  the  required  integral. 

(5.)  Supposing  ~  =  j9,  we  will  now  proceed  to  show  how 

to  integrate  the  equation  y  =  xp  +  F  (j)\  on  the  supposition 
that  F  {/j)  is  independent  of  x  and  y ;  noticing,  that  this 
equation  is  called  Olairaufs  form. 

Bj  differentiating  the  members  of  the  equation,  we  have 

dx      ^      ^         dx  dp     dx'' 

or,  erasing  the  terms  that  destroy  each  other,  we  have 


OF  THE   FIRST  ORDER  AND   HIGHER   DEGREES.        495 

Tliis  equation  is  satisfied  hy  putting  -J-  =  0  ov  dp  =  0, 

whose  integral  is  j9  =  C  =  const,  and  of  course  y  =  Cx  is 
the  proposed  equation. 

The  preceding  equation  can  also  be  satisfied  by  putting  its 
other  factor  equal  to  naught,  which  gives 

consequently,  if  -  is  a  function  of^;>,  by  finding^  from 

this,  and  substituting  its  value  in  the  proposed  equation,  we 
shall  get  an  equation  between  x  and  y,  which  does  not 
contain  any  arbitrary  constant,  and  is  hence  called  the  sin- 
gidar  solution  of  the  proposed  equation.     Thus,  if 

y  —  xp  +  af>^,     we  have     y  —  Cx  -\-  aOP' 

X 

for  the  integral,  or  j?  =  —  ^  =  0  is  the  singular  solution. 

Similarly,  if  y  =  px  -\-  a{\  +  p^\ 

we  shall  have  y  =  Cx  +  a{l  +  C-) 

X 

for  the  integral,  and  —  —  is  the  singular  solution. 

Eemarks. — 1.  If  we  have  the  equation  y  =^Vx  -\-  Q,  in 
which  P  and  Q  are  functions  of  j9,  then,  by  differentiating, 
we  shall  get 

dy      ^      xdV      dQ  .  d'P  dQ 


dx  dx         dx''  P  —  j9  P~j9* 

Taking  the  integral  of  this,  by  the  form  given  in  (4)  at 
p.  455,  we  have 


496  SINGULAR  SOLUTIONS  OF 

by  changing  y  and  dy  into  x  and  dx\  then,  eliminating 
p  from  this  and  the  proposed  equation,  we  shall  get  the 
integral  between  x  and  y. 

2.  It  is  manifest  from  what  has  been  done  in  the  first  part 
of  this  section,  that  in  the  a23plication  of  the  Differential  and 
Integral  Calculus  to  estimate  the  changes  of  position  of 
bodies,  which  result  from  the  violent  and  sudden  actions  of 
powerful  forces,  we  ought  generally  to  take  the  diflferentials 
of  the  variables  on  the  supposition  that  they  are  independent 
of  each  other,  since  the  tendency  of  the  actions  of  the  forces 
is  plainly  to  introduce  multiple  points  or  cusps  into  the 
motions  of  the  bodies.  Keciprocally,  in  finding  the  integrals 
of  differentials  thus  found,  we  ought  to  proceed  on  the  sup- 
position of  the  independence  of  the  variables,  as  explained 
at  p.  484,  &c. 

(6.)  From  what  has  been  done,  we  are  naturally  led  to  the 
consideration  of  what  are  called  the  Singular  Solutions  of 
Differential  Equations  of  the  First  Order. 

1.  If  F  (a?,  y,  c)  =:  0,  in  which  c  represents  a  constant, 
and  we  differentiate  the  equation,  regarding  c  alone  as  varia- 
ble, we  shall  have    ^^  ^^'  ^'  ""^  ^  =  0 ; 

dc  , 

then,  if,  as  in  the  example  at  p.  187,  we  eliminate  c  from  the 

■n  /           N        /^         ^     d¥  (x.  y,  c) 
equation    F  {x,  y,  c)  =  0    and     \J-MlU.  —  o, 

when  its  dimensions  exceed  the  first  degree,  the  result  will 
(generally)  be  what  is  called  a  singular  solution  of  the  pro- 
posed equation. 

2.  If  we  regard  x  and  y  as  being  functions  of  c,  then,  by 
differentiating  F  (a?,  y,c)=z  0  with  reference  to  c,  we  shall  have 


EQUATIONS   OF  THE   FIRST  ORDER.  497 

dx        do  dy        do  do  ' 

d^  (x,  V,  c) 
or,  since  V  ^'    ^  =  0, 

we  liave 

d'E  (a?,  y,  c)  dx  ^     ,    ^F  (x,  y.  c)  dy        ,         ^ 
dx         dc  dy         do 

Because  this  equation  must  evidently  be  satisfied  so  as  to 

leave  ^-_l1^),     or    ^'-Ii£l 

dx         ^  dy 

arbitrary,  we  must  have  either 

dx  __  dy  _ 

do  ~~    ''  do  ~~     "* 

wbicli  may  clearly  be  used  instead  of 

d^  (i^,  y>  c)  ^  Q 
do 

dv 
Similarly,  if  p  =  -—  ,  and  we  have  the  differential  equa- 
tion f  {x,  y,j))  =  0  such,  that  F  {x,  3/,  c)  =  0  represents  its 
singular  solution ;  then,  solving/"  (x,  y^  p)  =  0  relatively  to 
c,  we  get  the  form  c  ^^  0  (,t,  y,  p),  which  reduces 

F  {x,  y,c)  =  0     to  the  form     F  {x,  y,  &)  =  0, 

by  using  0  to  stand  for  the  function  d  (a?,  ?/,  p),  or  its  equiva- 
lent c. 

If,  for  brevity,  we  represent  the  first  member  of  this  equa- 
tion by  -w,  then,  since  the  function  0  may  contain  x  and  y, 
by  taking  the  differential  of  t^  =  0,  we  shall  have 

(dti.       du  do  dp\  Idu       du  dO  dp\    ,    _  ^ 

dx'^  Td'd^'dx}'^'''^  \dy  +  Wd^'Tyj'^y-^' 


498  SINGULAR  SOLUTIONS  OP 

which  must  clearly  be  satisfied  so  as  -to  leave  dx  and  dy 
arbitrary.  Hence,  we  may  clearly  put  the  coefficient  of  dx 
or  dy  equal  to  naught,  and  shall  thence  get 

dxi       du  dd  dp  _  du       du  dd  dp  _ 

dx'^  dd'd^'dx"    '     ^^    dy^  dd''dp'd~y~    ' 


ich  give 

dp  _       du   ^   du  dd 
dx~~       dx   '   dd' dp'' 

dp 
or    -f  = 
dy 

du 
dy 

du  dd 
""  dd'dp 

Because  d  is  used  for  d  (aj,  y,  p),  or  its  equivalent  c,  it  is 

{77/  fl^J 

clear  that  ~Ta^=^  ~t-  ^  which  (by  1),  for  the  singular  sol  ution, 
must  equal  naught ;  consequently,  because  —  =  0,  we  must 

have   -J-  =  infinity,   or  -^  =  infinity,  and  thence   —  =  0, 

di/ 
or  ^  =0,  which  are  the  conditions  for  finding  the  singular 

solutions  of  differential  equations  of  the  first  order.  Hence, 
if  p  is  eliminated  from  the  pyroposed  differential  equation 
hy  either  of  these  conditions^  and  if  the  result  satisfies  the 
proposed  differential  equation^  it  will  he  the  shigular  solu- 
tion of  it 

3.  To  simplify  t'le  applications  of  what  has  been  done 
as  much  as  possible,  we  shall  represent  the  proposed  differ- 
ential equation  f{x^  y^p)  —  0  hj  v^  which  gives,  supposing 
^  to  be  a  function  of  x  and  y, 

dv  ^        dv  ,         dv  dp  ^        dv  dp  ^         . 
dx  dy    ^       dp  dx  dp  dy    ^ 

and  thence  we  get 

dv  _        (dv       dv  dy\    ^    /dp       dp  dy\ 
dp  \dx       dy'^dxl    '    \dx       dy' dxr 


EQUATIO]SrS   OF  THE   FIRST   ORDER.  499 

which  reduces  -y-  to  nausrht,  since  -y-  or  —  in  the  divisor 
d_p  °  ax        ay 

is  infinite.     Hence,  —  ==  0  gives  the  values  of  p^  that  give 

the  singular  solution.  (See  Young,  p.  232,  &c.)  It  may 
be  noticed  if  i^  =  F  (a?,  y,  0)  =  0  does  not  contain  ?/,  tliat 
we  must  here  regard  x  as  being  a  function  of  y,  regarded 

as  being  the  independent  variable,  and  put  p'  =  -r-  for  p. 
(See  Young,  p.  237.) 

EXAMPLES. 

1.  To  find  the  singular  solution  of 

V  =  (x  -\-  y)p  —  xp^  —  (a  -{-  y)  =  0. 

dv 
Here  -y-  =  0  becomes  x  -\-  y  —  2x])  =  0,   which   gives 

^  ~      2x     '' 
which,  being  put  for^  in  the  given  equation,  gives 

Hence,  we  easily  get 

a?  —  y  =  2  \'ax^    or    y  =  a?  —  2  Vaa?, 

which  gives  -^  —  p  =  1 — ^ . 

ax  Yax 

From    y  =  X  —  2  Vax    we  get 
X  -[-  y  =  2x  —  2  Vax    and     a  +  y  =  a  +  x  —  2  Vax ; 

and  since    p  =^  1 -nr ,     we  thence  get 

Vax 

2{x-V^)  (l %^-xil  -  -%)^-  (a  +  aj-  V^)  = 

^         Vax^         ^         Vax' 

2  (i/a?  —  4/a)2  —  2  (|/aj  —  ^/aj  =  0 ; 


500  SINGULAR  SOLUTIONS  OF 

consequently,  since    y  =z  x  —2  Vax    satisfies  the  proposed 
equation,  it  must  be  its  singular  solution. 

2.  To  find  the  singular  solution  of 

V  =z  a?  +  2xyj[>  +  {a?  —  utr)  p'^  =  0. 

From     —  =  0    we  get    xy  -f-  {a^  —  x^)p  =  0  ] 
which,  multiplied  by  p  and  subtracted  from  the  proposed 
equation,  gives    x^  -\-  xyp  =  0    or    ^  = ; 

if 

which,  substituted  in  the  proposed  equation,  gives 

a.2  +  y2  _  ^2  ^  0 ; 

consequently,  since  this  satisfies  the  proposed  equation,  it 
must  be  its  singular  solution. 

3.  To  find  the  singular  solution  oi  xp  —  y  =  \/{x^  +  y'), 
or,  more  properly,  of  {xp  —  y)^  =z  x^  +  y\ 

Here  v  —  x^p^  —  %xyp  —  a?^  =  0, 

gives  -3-  =  0,  or  its  equivalent 


x'p  —  xy  =  0, 

or    p  = 

x' 

which  reduces  the  equation  to 

—  f  —  xi  —  0^     or     —  2/- 
as  required. 

4.  To  find  the  singular  solution  of 

=  ^, 

a;y  —  Ixyp  -f-  y""  - 

—  x^  —  p^ 

x''  =  0. 

Here  -7-  =  0  becomes 
dp 

^x'p  —  2xy  —  "Ix^p  —  0,     or     —  2ajy  =  ^> 
which  clearly  shows  that  the  question  does  not  admit  of  a 
singular  solution. 


E:.r^'ATI0>r3  OF  THE   FIRST  ORDER. 
5.  To  find  the  sinsfular  solution  of 


601 


V  =  (x'  —  2if)  iP-  —  ^xyp 


df 


0. 


dv 


Here    —  =  0     becomes     {oi?  —  2y^)^;>  —  2a?y  =  0, 

which,  multiplied  by  j?,  and  the  product  subtracted  from  the 
given  equation,  gives 


—  ^xyp  —  cc-  ==  0,     or    ^  =  — 


2y' 


consequently,  substituting  this  in  the  proposed  equation,  we 
have  (a?^  +  2y")  d^  =  0,  which  evidently  gives  x^  +  2y"  =  0 
for  the  singular  solution. 

6.    To  find  the  value  of  c,  which  gives  the  singular  solu- 
tion of  y  =  X  +  (g  —  If  {c  —  xf. 

Differentiating   the   equation  by   regarding    c    alone   as 
variable,  we  have 

2{G-l){c-xf  +  2{c-iy(c-x)  =  0,    or    c-x  +  g-1  =  0, 

which  gives  c  —  — ^ —  for  the  particular  solution,  since  the 

values  (?  =  1  and  c  =  a^,  which  also  satisfy  the  differential 
equation^  clearly  correspond  to  particular  integrals. 

T.  To  find  a  curve  such,  that  the  perpendiculars  drawn 


from  a  given  point  on  its  tangents  shall   be  constant,  or 
equal  to  each  other. 


502  SINGULAR  SOLUTIONS  OF 

Let  0  be  the  given  point,  taken  for  the  origin  of  the  co- 
ordinates, and  OP  for  the  perpendicular  from  the  given  point 
or  origin  to  the  given  line  AB ;  then,  we  may  clearly  suppose 

Y  —  y  =  -^  (X  —  a?),    or  its  equivalent    Y=:^X  +  y  —px^ 

to  be  the  equation  of  AB,  regarded  as  touching  a  circle 
having  OP  =  R  for  its  radius,  center  O,  and  Ojc  =  x,  Oij  =  y, 
for  the  rectangular  co-ordinates  of  the  point  of  contact  P  of 
the  tangent  and  circle. 

Supposing  y  to  decrease  when  x  increases,  we  shall  clearly 

have   —  -^  =z  —  p  for  the  tangent  of  the  angle  yOV,  and 

y  —px  equals  the  part  of  the  axis  of  y  between  AB  and  the 
origin  0.  Because  \]p^  -f  1  equals  the  secant  of  the  angle 
2/OP,  it  is  manifest  from  known  principles  of  trigonometry 
that  we  shall  have 

^  ~^^  =  R  =  const 
^f  +  1 

for  the  invariable  expression  of  the  perpendicular  from  the 
origin  to  the  tangents  to  the  sought  curve,  which  must  hence 
clearly  be  a  circle,  having  O  for  its  center  and  OP  =  R  for 
its  radius.  It  is  manifest  that  the  preceding  equation  may 
be  written  in  the  form 


y  =  j9a7  -1-  R  rj5>'^  +  1, 

an  equation  that  agrees  witb  Olairaut's  form  of  differential 
equations  given  at  p.  494,  whose  integral  is  there  shown  to  be 


y  —  ex  ■\-  ^V&  -^-1. 
The  singular  solution  of  this  gives 


c  = 


i/(R-^  -  af)' 


EQUATIONS   OF  THE   FIRST   ORDER, 

wliicli,  being  put  for  c,  we  have 


503 


y  = 


+ 


4/(R^-n 


and  thence  x^  -\-  qf  =  W 

is  the  singular  solution  of  the  proposed  question. 

8.  To  find  a  curve  such,  that  the  product  of  the  two  per- 
pendiculars from  two  given  points  on  any  tangent  shall  be 
constant  or  invariable. 


"^^^^^--^        y 

-^ 

r 

- 

^ 

D 

\                            c 

) 

F                             1 

3 

X 

Let  A  and  B  represent  the  given  points  through  which 
the  axis  of  x  is  supposed  to  pass,  the  origin  of  the  co-ordi- 
nates being  at  0,  the  middle  point  between  A  and  B ;  then 
we  may  suppose  that  the  sought  curve  touches  one  of  the 
tangents  CD  at  the  point  E  to  the  right  of  Oy,  the  axis  of  y, 
and  that  OF  and  FE  represent  the  x  and  y  which  corre- 
spond to  the  point  of  contact  of  the  curve  and  tangent. 

Representing  AO  =  OB  by  a  we  shall  have 

AF  =:  a  -\-  X    and     FB  =  a  —  a? ; 

then,  as  in  the  preceding  question,  Supposing  y  to  decrease 
when  X  increases,  we  shall  have  —  jp  —  the  tangent  of  the 
angle  of  inclination  of  CD  to  the  axis  of  a?,  and  thence 

AC  =  y  —  (a  +  a?) J?     and  BD  ^  y  -\-  {a  —  x)^] 


504:  SINGULAR  SOLUTIONS  OF 

consequently,  by  dividing  these  by  ^^{p^  +  1)>  we,  as  in  the 
preceding  example,  get 

y  -{a  +  x)p  y  +  {a  -  x)  p 

V{P'  +1)  V{f  +  1) 

for  the  perpendiculars  from  the  points  A  and  B  to  the  tangent 
CD ;  and  thence,  if  h^  denotes  their  product,  we  shall  have 

y  -  (^  +  ^)P  ^  y  -^  {^-x)p  _  T2. 
V{f  +1)  v'(i/  +  i)     ~    ' 

or,  by  performing  the  indicated  multiplication,  we  have 

y^  —  2pxy  —  y  {a^  —  0?°)  _ 
y  +  1  ~^ 

for  one  form  of  the  equation  of  the  sought  curve. 
Solving  the  equation  for  y,  we  readily  get 

y  =px  ±  \/(J/  4-  fn'p% 

in  which  m^  =  a^  +  h^;  this  being  a  differential  equation 
of  Clairaut's  form,  we  shall,  as  heretofore,  get 


y  =  Ox  ±  Vb'  +  m'G^ 
for  its  integral,  C  being  the  arbitrary  constant. 

By  taking  the  differential  of  this  equation,  supposing  C 
alone  to  be  variable,  we  readily  get 

for  the  singular  solution.     Taking  —  hx  for  ±  hx  in  C,  and 
•using  +  for  ±  in  y,  we  get  from  what  has  been  done 

my  +  h'^x^  =  m^h'^ 
for  the  equation  of  the  sought  curve,  when  the  perpendicu- 
lars to  the  tangents  do  not  fall  on  opposite  sides  of  the 
tangents,  and  it  is  manifest  that  the   curve  is  an  ellipse  ; 


EQUATIONS   OF   THE   FIRST   ORDER.  505 

noticing,  if  the  perpendiculars  are  drawn  on  opposite  sides 
of  the  tangents,  that  the  singular  solution  will  clearly  be 
an  hyperbola, 

9.  To  find  a  curve  such,  that  the  length  of  the  normal 
shall  be  a  (given)  function  of  the  distance  of  its  foot  from 
the  origin  of  the  abscissas.     (See  Lacroix,  p.  4:6Q.) 

Supposing  X  and  y  to  represent  the  co-ordinates  of  any 
point  of  the  proposed  curve,  it  is  easy  to  perceive  that 


represent  the  lengths  of  the  normal  and  the  distance  of  its 
foot  from  the  origin  of  the  co-ordinates ;  consequently, 


./m- !:=/(.+, I) 


will  express  the  differential  equation  of  the  question. 

It  is  easy  to  perceive  that  the  equation  {x  —  a)^+  y^  =  0, 
in  which  C  is  the  arbitrary  constant,  by  putting  C  =  /*  (of 
will  satisfy  the  question,  and  be  the  complete  integral,  since 
it  contains  the  arbitrary  constant  C.  For  by  taking  the  dif- 
ferential of  {x  —  of  +  y^  =:  C, 
we  have                  {x  —  a)  dx  -\-  ydy  =  0, 

which  gives     a^=  x  -\-  y  —^     and    x  —  a  ^=  —  y  -~^ 

so  that        f'£  +  f=fiaf=f(.  +  yf^; 

and  taking  the  square  root  of  the  members  of  this,  we  have 

22 


506  SINGULAR  SOLUTIONS  OF 

agreeing  with  tlie    assumed    differential   equation.      It  is 
manifest  that  (a?  —  a)-  +  3/^  =  c 

is  the  equation  of  a  circle,  the  axis  of  x  passing  through  its 
center,  a  being  the  abscissa  of  its  center,  and 
c  =f{af 

is  the  square  of  its  radius. 
If  we  take  the  differential  of 

{x-af  +  f  =  c=f{af, 

regarding  a  alone  as  variable,  we  shall  have 

then,  by  eliminating  a  from 

{:c-df  +  f=f{aY    and     -  {x- a)  =  f{a)f' {a), 

the  result  will  be  the  singular  solution  (called  by  Lacroix, 
the  particular  solution)  of  the  proposed  differential  equation. 
Kemarks. — 1st.  If  we  put 

c=^ka    in     (a?  —  of  -\- 1/  =  c^ 
it  will  become  {x  —  of  +  y^  =  ka, 

whose   differential  being  taken  by  regarding  a   alone   as 


variable,  is 

-{X-. 

«)  =  2' 

which  gives 

a  =  X 

*l. 

and  thence  the 

equation     {x 

-af  + 

f 

=  lea 

reduces  to 

7c' 

=  k(x 

+  2)' 

or     2/2 

= 

k(x-\- 

1). 

the  equation  of  a  parabola,  the  origin  of  the  co-ordinates 
being  at  the  focus  of  the  parabola ;  noticing,  that  this  is  a 


EQUATI02f3   OF   THE   FIRST   ORDER. 


507 


singular  solution,  compreb ended  under  the  general  singular 
solution  given  above. 
2d.  The  equations 

(x-df  +  f=f{a:f    and     _  (a,  -  a)  ^ /(«)/' (a), 
when  a  is  eliminated,  or  supposed  to  be  eliminated,  from 
them,  give,  when  taken  together,  a  result  which  is  some- 
times called  the  general  integral  of  the  differential  equation 


while       {x  —  of  +  if  —f{(if^ 

which  involves  the  arbitrary  function  f{af,  is  called  the 
complete  integral  of  the  same  equation.     Thus, 

xz^  —  yz  +  a  —  0     and     2x3  —  y  =  0, 
given  at  p.  187,  may  be  supposed  to  correspond  to 
(.^_„)^  +  y=  =/(«)=    and     -(x-a)=f{a)f{a); 

and  y"'  =  4aa?,  at  p.  187,  resulting  from  the  elimination  of  s, 
corresponds  to  the  elimination  of  a  from  the  preceding 
equations. 

10.  To  find  the  equation  of  a  curve  which  cuts  a  curve 
having  a  variable  parameter,  at  any  proposed  angle. 


Let  OB  represent  the  proposed  curve  when  referred  to  its 


608  SINGULAR  SOLUTIONS  OF 

rectangular  axes  as  in  the  figure,  and  ach  tlie  cutting  curve, 
when  referred  to  the  same  axes  ;  then,  if 

~-  •=  V'     and     -T-  =  p 
dx      -^  ax      -^ 

in  the  proposed  and  sought  curves,  stand  for  the  tangents 
which  the  tangents  to  their  arcs  at  their  point  c  of  inter- 
section make  with  the  axis  of  a?,  we  shall,  from  a  well- 
known  formula  of  trigonometry,  get 


tan  <p  = 


^-y'£ 


for  the  tangent  of  the  angle  which  the  sought  curve  makes 
with  the  proposed  curve  at  their  common  point  of  intersec- 
tion, which  is  supposed  to  be  a  given  angle  ;  consequently, 
representing  tan  0  by  a,  we  get 


dy 

dx      ^ 
.a  = 


-,       ,    and  thence  have   -/  =  ^- -. ; 

and  if  0  is  90°,  or  a  the  tangent  of  a  right  angle,  it  becomes 

infinite,  and  the  preceding  equation  reduces  to  -^  = -, . 

dx  jp 

Thus,  if  y  =  Ja?  is  the  equation  of  the  proposed  curve,  it 

gives 

dy 


h  z=zjp'  —  ^ ,     and  thence     -^  —  — 


x  dx       .  y 

X 


or  its  equivalent     {x  —  ay)  dy  —  {y  -{-  ax)  dx  =  0. 
Because  this  equation  is  reducible  to  the  form 


EQUATIONS   OF  THE   FIRST   ORDER.  509 

xdx  +  ydy 


4i) 


n^      I      ^,2         » 


1  +  iy}\     ""  ''^^-^y' 


bj  taking  the  integral,  and  using  a  log  C  for  the  arbitrary 
constant,  we  have 

tan-^  ^  =  ct  log  C  ^Qi?  +  2/1 

Because  Jog  represents  the  hyperbolic  logarithm,  if  A 
stands  for  the  base  of  a  system  of  logarithms  whose  modu- 
lus is  «,  by  writing  Log  for  a  log,  by  the  nature  of  loga- 
rithms, the  preceding  equation  may  be  written  in  the  form 

Log  h}-^^-^  ^  rrr  log  0  \^lF^y\ 

in  which  Log  denotes  a  logarithm  taken  in  the  system  whose 
base  is  A.     Putting 

tan~^  -  =  6    and    Vx^  +  ?/  —  r, 

on 

by  returning  from  the  logarithms  to  their  numbers,  we  shall 
get  A"  =  Cr  for  the  equation  of  the  sought  curve.  If  we 
suppose  7"  =  1  when  ^  =  0,  the  preceding  equation  gives 
C  =  1,  and  thence  the  preceding  equation  is  reduced  to 
A^  =:  ?',  which  is  clearly  the  logarithmic  spiral,  0  being  the 
constant  angle  at  which  the  radius  vector  i^  cuts  it,  and  A 
the  base  of  the  system  of  logarithms,  represented  by  it, 
a  =  tan  <f)  being  the  modulus.     (See  p.  136.) 

Eemarks. — 1.  If  0  =  90°,  a  =  tan  0  =:  infinity. 
2.  Hence  the  equation 

X  xdx  +  ydy 

=  a 


i  +  /i/y    ""2(2.^  +  2^) 


510  SINGULAR   SOLUTION'S   OP 

clearly  shows  that  we  must  have  xdx  +  ydy  =  0.  By 
taking  the  integral  of  this,  we  clearly  get  ar  +  y-  =  C, 
which  evidently  shows  the  sought  curve  to  be  a  circle,  C 
being  the  square  of  its  radius. 

For  another  example,  we  will  show  how  to  find  the  curve 
which  cuts  a  series  of  parabolas  whose  general  equation  is 
yti  _  g^m  r^^  riglit  angles. 

From  what  is  shown  at  p.  509,  we  must  take  -^  = , 

and  substitute  in  it  the  value  ^^  p'  ^-j- 1  as  determined 

from  the  equation  of  the  proposed  curve,  and  then  eliminate 
from  the  result  the  a  as  found  from  the  equation  of  the  pro- 
posed curve. 

mi  1  dy       w      ^'""^ 

Thus,  we  have        -^  =  —  a  — — r , 
ax       n      y'^~^ 

which,  put  for^',  reduces 

^  =  _1   to    <y  ^     ^  y""^  . 

dx  p'  dx  7nax'^~'^^ 

whose  members,  multiplied  by  the  corresponding  members 
of  the  given  equation,  we  get 

y'^dy  _        n  y'"~'^x^ 
dx    '~        m    ij?'"-^    ' 

or  its  equivalent       'mydy  +  nxdx  =  0 ; 

whose  integral  is  clearly  the  ellipse 

my'^  +  nx^  =  C, 

in  which  the  constant  C  represents  the  product  of  the 
squares  of  the  semi-axes  of  the  ellipse. 


SECTION  YIIL 

INTEaRATION  OF  DIFFERENTIAL  EQUATIONS   OF  THE  SECOND 
AND   HIGHER   ORDERS,  BETWEEN  TWO   VARIABLES. 

(1.)  The  most  general  form  of  a  differential  equation  of 
the  second  order  between  two  variables,  may  evidently  be 

supposed  to  be  reduced  so  as  to  contain  a?,  2/,  ;t-  ,  y^ '   ^^' 
gether  witli  constants. 

(2.)  Supposing  'W  =  0  to  be  the  primitive  or  integral  of 
tbe  second  order  of  the  preceding  differential  equation,  con- 
taining the  two  arbitrary  constants,  C  and  C\  which  have 
clearly  resulted  from  two  successive  integrations  of  the  pro- 
posed equation,  then,  by  taking  the  first  and  second  dif- 
ferentials of  u  =  0,  regarding  x  as  being  the  independent 
variable,  we  shall  have  the  three  equations  ^^  =  0,  du=z  0, 
d^u:=zO;  consequently,  eliminating  C  and  C  from  these 
equations,  we  shall  clearly  obtain  the  proposed  equation  of 

the  second  order  between  a?,  y,  ~ ,  -~ ,  which  is  clearly 

independent  of  the  constants  C  and  C^  Now,  if  we  elimi- 
nate C^  from  u  =  0  and  du  =  0,  we  shall  get  a  differential 
equation  of  the  first  order  denoted  by  Y  =  0,  involving  the 
constant  C ;  and,  in  like  manner,  by  eliminating  C  from 
u  —  O  and  du  —  0,  we  shall  get  an  equation  of  the  first  order 
denoted  by  V  =  0,  involving  the  constant  0^  It  is  easy  to 
perceive  that  if  we  eliminate  C  between  Y  =  0  and  dY  =  0, 
or  eliminate   G'  between  Y'  =  0   and  dY^  =  0,  we  shall 


612  DIFFERENTIAL  EQUATIONS  OF  THE 

obtain  the  proposed  differential  of  tlie  second  order,  so  tliat 

V  =  0  and  V  =  0  will  eacli  be  a  differential  equation  of  the 

first  order  of  the  proposed  equation  of  the  second  order. 

dy 
Hence,  if  we  eliminate  3-  from  Y  =  0  and  Y'  =  0,  it  is 

clear  that  the  result  will  he  u  =  0,  the  primitive  of  the 
pi'oposed  differential  equation  of  the  second  order.  (See 
Lacroix,  pp.  292  and  293.) 

Thus,  supposing  y  ■\-  ax  +  h=^0  to  represent  the  primi- 
tive (or  second)  integral  of  a  differential  equation  of  the 
second  order,  having  a  and  h  for  its  arbitrary  constants,  then, 

by  differentiating  it,  we  get  -^  -f-  a  =  0,  which  is  said  to  be 

obtained  from  the  primitive  by  a  direct  differentiation.    But 

if  we  divide  the  primitive  by  a?,  we  get (-  a  =  0, 

whose  differential  gives 

xdy  -iy  +  l)dx=zO,     or     y  -  a?  ^  +  J  =  0, 

which  is  said  to  be  obtained  from  the  primitive  by  an  in^ 
direct  differentiation. 

Thus  we  have  obtained  the  differential  equations 

-^  +  a  =  0     and     y  —  x~-^h  =  0, 
dx  ^  dx 

which  are  both  of  the  first  order,  the  first  containing  the  con- 
stant a,  and  the  second  the  constant  h.     If  we  differentiate 

d'lj 
these  equations,  they  will  concur  in  giving  -j~  =  0  for  the 

differential  equation  of  the  second  order,  of  which  the  two 
preceding  equations  will  be  the  first  integrals ;   and  if  we 

eliminate  -~  from 
dx 


SECOND  AND  HIGHER  ORDEKS.  613 

dy  dv 

-^j^a  —  0    and    y— a? -^- +5  =  0,    we  get    y-\-ax-\-'b  =  0^ 

wMch  is  the  primitive  or  second  integral  of  —~  =  0. 

In  like  manner,  it  is  clear  that  a  differential  equation  of 
the  third  order  has  three  differential  equations  of  the  second 
order  for  its  first  integrals,  &c. ;  and  so  on,  to  any  extent. 

(3.)  We  now  propose  to  show  that  any  differential  equa- 
tion between  two  variables,  has  an  integral,  such,  that  the 
nth.  order,  in  its  most  general  form,  contains  n  arbitrary- 
constants. 

For  conceiving  the  equation  to  be  solved  with  respect  to 
the  differential  coefficient  of  the  n\h  order,  we  shall  clearly 


get  -~^  =  a  function  of 

dy      d-y                  d^-'^y 
^'     ^'     dx'     dx^'    '  '  '  '  dx^'-'' 

which,  by  successive  differentiations,  gives 

-j-r-A  =     a  function  of    a?,     y,     -f,     -^,    .  . 
dx^  +  ^                                     '     •^'     dx'     dixr' 

•  •  •  dx^' 

d'^'-'y             .      ,.         .                   dy      dy 

and  so  on,  to  any  required  extent.     It  is  hence  easy  to  per- 
ceive that,  by  obvious  substitutions, 

dy       d-^y       d-^'y 

dx^'      ^^n  +  l)      c^^»  +  2»      <^c., 

are  all  known  functions  of 

dy       d^y  c?"  -  ^y 

which  clearly  (generally)  consists  of  n  terms. 

22* 


514  DIFFERENTIAL  EQUATIONS  OF  THE 

From  Taylor's  theorem  (see  p.  15),  we  shall  have 

consequently,  if  x^  is  such  a  value  of  x  as  does  not  make 

dy       dry  d'-^y 

^'     dx'     dx''     dx''-^' 

in  the  preceding  differential  equation,  any  of  them,  infinite ; 
then,  supposing  x^  to  be  substituted  for  x  in 

dy       d^y  d'^~^y 

y^  Tx'     ^'     rZ^^' 

and  that  A,  Ai ,  Aj , A„  _  i ,  represent  their  result- 
ing values,  if  we  change  h  into  x  —  x^  in  the  preceding 
expansion,  and  change  y'  into  y  after  the  substitutions  of 

the  values  of     y,%%, JJ, 

we  shall  have 

y  =  A  +  A'(a.-^0  +  A,^-^^-^V 

1.2 (^i  -  1)  "^  afiz;'^    1.2 n'^'  ^'' 

for  the  integral  of  the  proposed  differential  equation,  whose 
first  n  terms  clearly  contain  A,  A^,  &;c.,  for  the  n  arbitrary 
constants  required. 

(4.)  Any  diffei-ential  equation  of  the  wth  order  between 
X  and  y  must  have  n  differentials  of  the  order  n  —  1  for  its 
first  inteprals. 

For,  as  in  the  preceding  proposition,  Taylor's  theorem 
,    dy  ,        d'y  K"         d?y    h^ 

which,  by  changing  A  into  —  x^  and  representing  the  value 


SECOND   AND   HIGHER   ORDERS.  515 

of  y',  correspondiDg  to  a?  =  0  by  A,  becomes 

If  in  this  we  change  A  into  Ai,  As,  &c.,  y  into  -^,  -y^  » 

&c.,  we  shall,  in  like  manner,  get 

.    _  dy       (Py  d^y  x^        d^y     x^  . 

^'~^~^^'^^i:2~5?IX3"^'  ^'^•' 

^'  "■  ^^^       ^2^  1  "^  6/a;^  1.2      '        ' 
and  so  on,  until  we  obtain  n  —  1  equations.     If  we  now 
eliminate  from  the  n  equations  which  we  have  obtained, 
^ny       dJ'^^y      d^'^hj 

and  so  on ;  the  results  will  clearly  be  the  first  or  {n  —  l)th 
integrals  of  the  proposed  differential  of  the  ??th  order,  as 
required,  which  have  A,  Ai,  A2,  &c.,  for  their  arbitrary 
constants. 

Thus,  if  71  =  2,  the  first  integrals  are 

^  =  2^  -  i*  +/(^'  y^  2)  S  -/'  (-'  y'  1)  li  +'  ^'-^ 

and 

^'=1  -/(^'  y^  i)  i  +/'  (^'  2^'  i)  fi  -•  ^°- 

(See  Lacroix,  p.  297.) 

(5.)  We  now  propose  to  show* how  to  find  the  integrals 
of  differential  equations  of  the  second  order  between  x  and 

7/,  when,  besides  -~ ,  they  contain  x  or  y,  or  -j^ ,  or  when, 

besides  -j4. ,  they  contain  x  and  -j  or  y  and  -^  ,  by  the 
common  methods.     (See  Young,  p.  243,  &c.) 


616  DIFFERENTIAL   EQUATIONS  OF  THE 

1.  To  integrate  a  differential  equation  of  the  form 

72 

Solving  the  equation  with  reference  to  -7^ ,  we  evidently 
get  -T^  =  X  =  a  function  of  a?, 

which,  multiplied  by  dx  and  integrated,  gives 

and  this  multiplied  by  dx  and  integrated  again,  gives 
y  =  fdxfxdx  +  C^  +  C  =    Pxda^  +  Cx-\-  0\ 
Thus,  if  X  =  x%  we  have 

//»2                                       /v,n  +  2 
X'dx''  =    /    X-dx'  =  J- =V7 ^^  > 
J                  (7J,+  1)  {n  +  2)' 

a^d  thence      ^  =  ^_^1__  +  C^  +  C^ 

2.  To  integrate  a  differential  equation  of  the  form 

d^'yy 


='('.S)  =  «- 


Here  we  have  -^-^  =  Y,  which,  by  putting 
-^  =  ^    becomes    -^  =  Y, 
whose  members  multiplied  by 

P=.£,    become    p£  =  Y£, 

whose  members  multiplied  by  dx  and  then  integi'ated,  give 

^  =  fYdy-{-C    of  the  form    "I'^Y'+C. 


SECOND  AND   HIGHER  ORDERS.  517 

From  this  we  immediately  get  —^  =  KjyT'Tl'u  i  ^^^  thence 

1       dx  1  r  dy 

—       -  or    a?—   ' 


=/ 


p~dy~dy~  ^"iij'  +  Q)  J    |/(2Y^  +  2C)* 

dx 
which  is  of  an  integrable  form. 

Thus,  if  Y  =  —  -„ ,  we  shall  have 

^  --J  -a^---2a^^^^ 
and  thence  we  readily  get  the  form 


/ady        _  ^    r  ^y 


noticing,  that  C^  has  been  used  for  the  first  arbitrary  con- 
stant. 

3.  To  find  the  integral  of  the  form  ^ (^, ^)  =  0, 

or  of  its  equivalent  F  f^,  -^  J   =0. 

The  equation  solved  for  ~ ,  gives  -~  =Y  =  £l  function  of 
CvX  ccx 

p,  and  then  dx  =  -~-j  an  integrable  form,  whose  integral  will 


be  expressed  hy  x=  /  p  5  ^^^j  since  ;t^  =i>,  we  also  have 

'Ddj) 
dy  =  _pdx  =-—- J  an  integrable  form,  whose  integral  will  be 

expressed  by  the  form  y  —  f-^  •     Hence,  eliminating  p 


518  DIFFERENTIAL  EQUATIONS  OF  THE 

from  the  equations 

.  =  /f     ana    y  =  f^l, 

we  shall  get  the  required  relation  between  x  and  y.     Thus,  if 

we  easily  get  a-~  =  dx^ 

whose  integral  gives 

C  —  ajp-'^zzzx    or    p  = 


and  thence,  since  dy  =  pdx,  we  have  y^=(i  log  C  (C  —  a?) 
C  and  a  log  C  being  the  arbitrary  constants. 

4.  To  integrate  the  form  F  U  ^ ,  2)  =0, 

or  its  equivalent,  F  lx,2?,  -j-\=0. 

This  being  an  equation  between  two  variables,  x  and  p,  by 
regarding  p  as  being  a  function  of  x  taken  for  the  inde- 
pendent variable,  may  be  integrated  by  the  methods 
given  in  Section  V.,  which  will  give  the  form  F  (a?,  p^c)  =  0; 
in  which  c  denotes  the  arbitrary  constant. 

If  we  can  from  this  equation  find  j9,  we  shall  get 
^  =  X  =  a  function  of  a?, 

and  thence^  =  -^  gives  dy  =  'X.dx  or  y  =    I  X.dx,  whose 

integral  gives  the  relation  between  x  and  y. 

But  if  we  can  not  find  j9  in  a  function  of  x,  or  can  find  x 
more  readily,  then  we  shall  get  the  form  a?  =  P  =  a  func- 
tion of  p,  whose  differential  gives  dx  =  dF ;  and  thence 
from   dy  =  pdx  we  shall  get  dy  =  pdPj  whose  integral 


SECOND  AND  HIGHER  ORDERS.  519 

y  =    j  pdF  gives  the  relation  between  y  and  ^.     Hence, 

from  the  elimination  of  ^  from  a?  =  P  and  y  =    I  pdF,  we 
shall  get  the  relation  between  x  and  y. 

If  F  (a?,  j9,  c)  can  not  be  readily  solved  for^  or  x,  we  may 

put  -~  for^,  and  then  try  to  integrate  the  result,  by  the 

methods  for  integrating  differential  equations  bstween  two 
variables. 


Thus  ^^^(l+^T 

^'        ,.  2xdx'       V  ^  dxV   ' 

or  its  equivalent        2xdx  = ^-— t» 

{i+rf 

lias       .  «=  +  C  =       "'-^     , 

for  its  integral ;  and  by  subtracting  the  squares  of  the  mem- 
bers of  this  from  a*,  we  have 

«^-(^+C/  =  ^-„    or    Va^_(^  +  0)---jl^^, 

consequently,  dividing  the  members  of 

^=  +  C  =  --^1= 

by  the  corresponding  members  of  the  preceding  equation, 

we  get  p  =  ,    and  thence 


{a^  4-  0/ 
an  integrable  form. 


or  2/ 


620  DIFFERENTIAL  EQUATIONS  OF  THE 

For  another  example,  let  there  be  taken  the  equation 
or  its  equivalent 

Multiplying  the  members  of  this  by  dx  and  dividing  the 
products  by  1  +  jr^  we  readily  get 

dx  4-  -,     ,      2  xdp  =  -— ^ dr>, 

which  is*  a  linear  equation,  agreeing  with  the  equation  at  p. 
455,  when  we  put  x  for  y,  ai^d^  for  x,  and  change  P  and  Q 

into  rr-^ — «    and 


Hence,  from  the  integral  given  at  p.  455,  we  have 
X  =  6~-^iTp^  (a   /  ^^r^^ ^- —  +  C) 

\    ^  VlH-y  / 

Because 

1 


the  value  of  x  is  clearly  equivalent  to 

By  taking  the  differential  of  this,  we  have 
which,  since  dy  =^dx^  gives 


SECOND   AND   HIGHER   ORDERS.  521 


,   _    cijpdp         Gp'^dp  _    apdp  dip  Qdp 

whose  integral  gives 


y  = •  A ==  —  C  loo;^^^^ -pr-, — =^ 


Op  —  a        ^  ,      p  +  Vi  +  p^ 

noticing,  that  we  here  use  C  log  ;^,  for  the  arbitrary  constant. 

By  eliminating  p  from  a?  and  y,  the  equation  between  x  and 
2/  will  be  found  as  required. 

For  another  example,  we  will  find  the  integral  of 


2  la'  "^  ^J\^  -x^ 


or  its  equivalent      2  {a^p'  +  i»")  <^  =  xpdXy 
which  is  clearly  homogeneous  in  p  and  x. 

By  putting  x  =pz^   the  equation  is   easily  reduced  to 

2{a''  +  z')dp=^B{zdp-{-pd3\     or     -£  =  ^~-^, 

whose  integral  is 

log^  =  log  0  V(2a2  ^  ^2)^     Q^    ^^  =r  C  |/(2a'  +  s') ; 
consequently,  from  x=zp3  we  have 

From  x=pz  we  have 
and  thence,  from  cZ?/  =  fdx^  we  have 


622  DIFFERENTIAL  EQUATIONS  OF  THE 

and  from  x  =  Cs  ^  (2a^  +  z'), 

which  gives  o^a?  =  C  -/  (2a'  +  ^)  dz  +      .^  "^^    ""  ^.  , 

we  get  (^y  =  C  (2a'  6^2  +  2^-  dz\ 

2 
whose  integral  is    y  =  -  C^  z{Za?  +  2-)  +  Q'. 

o 

If  we  now  eliminate  z  from  the  values  of  x  and  y,  we  shall 
get  the  sought  equation  between  x  and  y,  as  required. 

5.  To  integrate  an  equation  of  the  form 


^  V'  dx'  dx^I  ~  ^' 


or  its  equivalent         F  |y,  ^,  -j-j 


Because  dx  =  —  ,  we  have  the  form  reduced  to 


F(..,f)  =  0; 


dy 

and  it  is  clear  that  we  may  now  proceed  in  much  the  same 
way  as  before.     Thus,  to  integrate 

or  its  equivalent 

By  putting  ^-  for  -^ ,  the  equation  is  easily  reduced  to 

{^  +  yP)  £.  =  ^  +  V\     or     {a  +  yp)  ^^  =  (1  +  f)  dy, 
which  gives  the  linear  equation 

^y  -  YT^y^P  =  xTf^P- 


SECOND   AKD   HIGHEE   ORDERS.  523 

Comparing  this  to  tlie  linear  equation  at  p.  455,  we  have 

and  for  dx  we  have  dj) ;  consequently,  we  shall,  from  the 
formula  at  p.  455,  get 


r  Pdp     /         /»     /•  —pdp  -^  \ 


or  since 


4/(1  +y)  ='6'°g  ^^  +  ^'^     and      ^^ =  ^-^°^  '^'^^  +  P% 

r  1  +  jp^ 

we  shall  have      j/  =  aj?  +  C  ^/(l  +  i>^) ; 
and  from  dx  =  ~     we  shall  get 

J9  1/(1  +i>')' 

whose  integral  is 


=  a  log^  4-  C  log  Q'  {p  +  Vl  +  y) 


=:  log  [^«  (C>  +  C'  ^1  +  i>0  ^• 
Eliminating^  from  a?  and  y,  we  shall  get  the  sought  equa- 
tion between  x  and  y. 

6.  When  a  differential  equation  between  a?  and  y\s  of  the 
second  order,  and  x  is  taken  for  the  independent  variable, 
then,  regarding  dx^  dy^  and  d'y  each  as  being  variables  of 
one  dimension,  when  the  proposed  equation  is  homogeneous 
in  terms  of  its  variables  and  their  differentials,  it  can  be  re- 
duced to  a  differential  of  the  first  order,  which  does  not 

contain  »,  by  putting  y  =  -ua?  and  — (  =  ^  in  it.     For  if  n 

dx         sa 


524  DIFFERENTIAL  EQUATIONS  OF  THE 

denotes  the  degree  of  homogeneity  of  the  equation,  it  is 

plain,  since  -t4  is  clearly  of  —  1  dimensions,  that  it  must 

have  a  factor  of  ?i  4-  1  dimensions ;  and  since  vx  is  put  for 
y,  it  is  evident  that  wherever  y  is,  a?'*  must  enter  as  a  factor ; 

it  is  also  evident,  since  -^  is  of  no  dimensions,  that  it  must 
have  a?"  for  a  factor. 

Because  the  remaining  terms  of  the  equation  are  of  n 
dimensions,  it  is  manifest  that  after  the  preceding  substi- 
tutions we  may  divide  the  equation  by  a?**,  and  thus  free  it 
of  X. 

Thus,  to  find  the  integral  of 

xd^y  =  dydx^ 
we  divide  its  members  by  dx^^  and  thence  get 

da?        dx^ 

in  wliich  we  put    -~  =  -    and    -^  =  p. 
^        ditr       X  dx      -^^ 

and  thence  get  q  =  j[)-^ 

and  since     -^  =  ^    is  the  same  as    -f-  =  ^      by  putting 
dx-        X  dx       x^       ''  ^         ° 

p  for  ^,  we  thence  get 

dp  _p  dp  _  dx  ^ 

dx~  x^  p   ~    X  ^ 

and  from  y  =  vx  we  have 

,  ,  ,  ,  dx         dv 

dy  =  pdx  =  vdx  +  xdv,     or     —  = , 

^      ^  '  X       p  —  v^ 

and  hence  get 

dp  dv  ,  ^  , 

'^'=JZr-^^     ^^    Fdp  =  pdv  +  vdp', 


SECOND   AND   HIGHER   ORDERS.  525 

and  taking  the  integral  of  this,  we  have 

-|^=^?)  +  C,     or    /  =  2p?;  +  2C. 

cloa        (I'D 
Also,  by  taking   the  integrals   of    —  =  — ,    we  have 

log  X  =  log  pG\      or      a?  =  ^C\     which  gives     i?  ==  p7 . 
Substituting  this  value  of  p  in  ^^  =  2jpv  +  20,    we  get 

x'  ^  2G'xv  +  2CC^^ 
which,  since  y  =  xv,  may  clearly  be  represented  by 
x'  =:  2Cy  +  C; 

when  C  and  C^  represent  the  arbitrary  constants. 
Otherwise. — Since  the  equation 

a?  -r^,  =  -^     IS  reducible  to     —  =  -7— , 
ddir       ax  X        dy  ' 

dx 

by  taking  the  integral  we  have 

log  a;  =  log  C  ~ ,     or    x  =  C  ~,     or    xdx  =  Qdy^ 

whose  integral    '—  =  Cy  +  const,     gives    o?  ~  2Qy  -f  C, 
the  same  as  by  the  preceding  process.     Again,  if 

xd?y  =  ady^  +  hdx^j     or     x  -—  =  a  -—,  +  5, 

then,  putting  ^  and^  for  — ^  and  -^,  we  have  <i  =  aj^"^  -\-  h, 

-o  dp       q  dx       dp 

Because  -f-  =  ^ ,     or    —  =  -^  , 

ax       X  X  q  ^ 

by  substituting  the  value  of  </,  we  shall  have 


626  DIFFERENTIAL  EQUATIONS  OF  THE 


dx  _  dp       _  dp 

X   ~  ajp^  -f  h 


(l+^y)       V-ab(l+f'^^ 


whose  integral  gives 


or  we  have  Qx  =  6*'"*  , 


JLtan-^P|/? 


and  thence  a?  =  ^  ^ 

in  which  e  is  the  hyperbolic  base,  and  C  is  an  arbitrary 
constant. 

Since  dy  =  pdx.    from    —  =  — „      , 

^       -^      '  a?        ajp^  ■\-h 

we  easily  get     c?y  =  ^"^  ><  ^' 

jind  thence,  from  the  substitution  of  the  value  of  a?,  we  get 

and  integrating  this,  we  get  y  in  a  function  of  ^,  with  a  new 
arbitrary  constant.  Hence,  eliminating  jp  from  the  values 
of  X  and  y,  the  equation  between  x  and  y  will  be  determined 
as  required. 

7.  Finally,  it  may  be  added  that  there  are  two  classes  of 
equations  of  the  forms 

in  which  n  is  supposed  to  be  a  positive  integer  greater  than 
2 ;  such  that  they  can  be  reduced  to  the  forms  of  equations 


SECOND   AND   HIGHER   ORDERS.  527 

of  the  first  and  second  degrees,  which  we  have  heretofore 
seen  how  to  integrate. 

For,  bj  putting     -i-~i  —  'W,    in  the  first  of  the  preceding 
equations,  it  reduces  to     F  l-y-  ,    wl  =:  0,     which  is  clearly 

of  the  form  of  a  differential  equation  of  the  first  order  be- 
tween u  and  a?,  which  gives  w  =  X  =  a  function  of  a?,  and 

/«  —  1 
Xc?a?"  -  ^ 

To  integrate  the  second  of  the  preceding  equations,  we  put 

^-  =  «,     and  thence  get     ;^„  =  ^. ; 
and,  by  substitution,  the  equation  becomes 


"©.•l-o. 


-2 


an  equation  of  the  second  order,  which  we  have  seen  how  to 
integrate  at  p.  516.  Hence,  we  shall  have  'w  =  X  =  a 
function  of  x ;  or,  restoring  the  value  of  ?/,  we  have 

^n  — 2,w  /»«  — 2 

-j-~i  =  X,     which  gives     y  =    I       X-dx""- 

Thus,  to  integrate  -7^.-7^3  —  1,  we  put  -^  =:  r,  and  thence 

dr 
get  r  -J-  —  1,  or  rdr  =  dx,  for  the  transformed  equation. 

By  taking  the  integral   of  the   preceding   equation,  we 

ffi  

have  —  +  C  —  a',     or    r  =  r  2  (-«  —  C). 

Denoting  the  differential  coefficient  of  the  next  inferior 
order  to  r  by  q,  we  shall  clearly  have 

dq  =  rdx  =  r^dr^     or     q  =  —  -{-  Cij 


528  DIFFERENTIAL  EQUATIONS  OF  THE 


and  since  dp  =  qdx  =  l~  +  CA  rdr^ 

or  p  = 

consequently,  from 


^  =  3:5  +  0  +  ^- 


dy  =  pdx  =  (~^  +  172  +  ^)  ^^^» 

^^g^*         ^  =  3:5:7  +  0:1  +  1:2  +  ^ 
■  for  the  complete  integral,  in  which  r  =  V2  {x  —  C). 

For  an  example  of  an  integral  of  the  second  order,  we 

will  take  -7^  =  -7^. 

ax*       dx' 

Puttmg      q  =  ^,    we  have    ^  =  ^„ 

and  thence  the  equation  is  reduced  to 

d'q 

d^  =  ^'^ 
whose  members,  multiplied  by  dq^  give 

%  =  q^  +  0\     or     dx=-M=^,, 
dxr       ^  4/^2 +  C« 

whose  integral  is    x  =  log  ^- a~, , 

in  which  log  p^,  is  1 
integration.     From 


in  which  log  p>  is  the  arbitrary  constant  introduced  by  this 


q  =  i~7,    and     dx  =  — — ^ — ? 
dif  ^^-2  ^  0* 

we  easily  get  dp  =  qrljo  —       ^  ^ — , 

and  by  taking  the  integral 

i>  =  V^r  -i  C"-  +  C     and     dy  -  jxle  —  dq  ^  O'dx, 


SECOND   AND    HIGHER   ORDERS.  529 

whose  integral  is       y  z=:  q  -\-  Q'x  -f  C ; 
e  being  the  hyperbolic  base, 

aj  =  log  ^^±i^-l+—  is  equivalent  to  Q'e' =  q-\-V^G\ 
which  gives 

(CV-^y  =  /  +  C^     or    CV^-2CV^=:C^ 

and  thence  q  = ^r^^j  e-\ 

From  the  substitution  of  ^  in  y  =  q  ■\- Q,'  x  '\-  Q,'\  we  have 

which  may  more  readily  be  represented  by 

y  =  Ce^  +  Ci6-^  +  C^a?  +  Cs, 

C,  Ci ,  Ca ,  and  Cg  being  the  arbitrary  constants. 

(6.)  We  will  now  show  how.  to  lind  the  integral  of  a  linear 
eqtiation  of  the  nth  order,  represented  by 

;s^+^^  +  B^  + +  Mj  +  %  +  x  =  o, 

in  which  the  coefficients  A,  B,  C,  &c.,  may  either  be  constants, 
or  they  may  contain  the  independent  variable  x  without  y. 
We  shall  determine  the  general  form  of  the  sought  integral 
by  integrating  the  more  simple  equation 

which  is  independent  of  X,  and  the  coefficients  are  supposed 
to  be  constants. 

Supposing  e  to  be  the  base  of  hyperbolic  logarithms,  C 
and  m  constants  to  be  determined;  if  we  represent  y  by 
Q^mx  ^Q^  ^j-ig  j^g  ^^  p^  gg^  ^g  g]^g^21  have 

23 


630  DIFFEREKTIAL  EQUATIONS   OF   THE 

dy  =  Cde"'''  =  mCe"'''dx,     or    -/  =  mCe""', 

CIS} 

and  in  like  manner 

dor  ^    dsc^  ' 

and  so  on,  to  any  required  extent.     Hence,  by  substituting 

the  values  of      g,   J;^,    g:^,    &c., 

in  the  preceding  equation,  and  rejecting  the  factors  C  and 
^mx^  whicb  will  be  common  to  all  the  terms  of  the  resulting 
equation,  we  shall  get  the  algebraic  equation  of  the  nth. 
degree, 

m"  +  Am"-^  +  Bm""-^  + +  Mm  +  N  =  0, 

which,  from  the  well-known  theory  of  equations,  must  have  n 
roots.  Solving  the  equation,  we  shall  have  the  n  roots,  which 
may  be  denoted  by  mj,  m^,  Wg,  and  so  on,  to  7?i„  inclusive  ; 
and  since  each  of  these  roots  satisfies  the  equation,  if  Ci,  Ca, 
Cg,  and  so  on,  to  C„  inclusive,  denote  the  const.,  then,  if  the 
roots  are  unequal,  we  shall  clearly  have 

for  the  complete  integral  of  the  proposed  equation ;  as  is 
manifest  from  the  circumstance  that  each  of  its  terms  satis- 
fies it,  and  of  course  all  its  terms,  conjunctly,  will  satisfy  it; 
and,  since  y  contains  n  constants,  or  a  number  equal  to  the 
order  of  the  proposed  differential  equation,  it  evidently  re- 
sults that  the  preceding  equation  between  x  and  y  must  be 
the  complete  integral  required.  It  may  be  added,  that,  so 
long  as  the  roots  of  the  equation  are  unequal,  the  integral 
will  be  of  the  preceding  form,  whether  the  roots  are  all  real, 
or  some  (an  even  number)  of  them  are  imaginary.     If  two 


SECOi^D   AND   HIGIIEB   ORDEKS.  631 

of  the  roots  of  the  equation,  as  m^  and  ^/i^,  are  imaginary, 
they  may  be  expressed  by  the  forms 

a  +  h  V"—l     and     a  —  h  V~\ 

since  (see  p.  440  of  my  Algebra)  the  imaginary  roots  of 
equations  are  known  to  occur  in  pairs  of  the  preceding 
forms  ;  consequently,  for  the  terms  Ci6"'i'"  +  Csc'^a^  of  y,  we 
may  write 

and  since  (see  p.  58), 

^bxV-i  _  ^Qg  J)x-\-  V  —  1  sin  hx 

and  g-6x  V  _  1  _  ^^g  })x—  V  —1  sin  hx, 

this  is  easily  reduced  to 

e^^  [(Ci  +  C2)  cos  bx  +  (Ci  -  C2)  4/"^!  sin  hx']. 

Since  we  may  clearly  assume  such  values  for  the  constants 
Ci  and  Co,  that  we  shall  have 

Ci  +  0.2=^  sin  q     and     (C^  —  C^)  V  —1  =p  cos  ^, 

we  shall  thence  change  the  preceding  expression  into 

jpe"'"'  sin  (l)x  -\-  q). 

Hence  it  results  that  we  have  reduced  the  terms 
Ci6'"i="  +  C,6^^^  of  y     to  the  form    pe"""  sin  (bx  +  q\ 

and  it  is  clear  that  every  other  corresponding  pair  of  im- 
aginary roots  will  admit  of  like  reductions. 

It  may  be  noticed,  that  if  our  equation  has  two  or  more 
equal  roots,  our  method  of  finding  the  equation  between  x 
and  y  will  fail  for  the  equal  roots,  and  will  be  applicable 
only  to  the  unequal  roots.     Thus,  if  in  the  equation  between 


632  DIFFERENTIAL   EQUATIONS   OF  THE 

X  and  y  we  suppose  the  root  mj  equals  the  root  7/23,  the 
equation  will  become 

which  clearly  shows  that  the  two  constants  Ci  and  Co  are 
actually  only  equivalent  to  a  single  constant  represented  by 
Ci  +  C2,  so  that  the  equation,  will  be  defective  in  not  having 
a  sufficient  number  of  constants,  which  ought  to  equal  n^  the 
order  of  the  proposed  differential  equation. 

The  defect  may  easily  be  remedied  by  writing  the  terms 
containing  the  equal  roots  in  the  form 

if  three  roots,  as  m^,  m^,  mg,  are  equal,  they  must  be 
written  in  the  form 

and  so  on ;  observing,  that  the  index  of  x  in  the  last  of  the 
terms  of  this  kind  is  less  by  a  unit  than  the  number  of  equal 
roots.  To  be  satisfied  as  to  the  correctness  of  what  has  been 
said,  it  will  be  sufficient  to  apply  it  to  the  integral  of  the 

equation        g  +  A  g  +  B  |  +  Cy  =  0, 

which  is  clearly  a  particular  case  of  the  general  differential 
equation,  obtained  from  it  by  putting  3  for  n  in  it ;  conse- 
quently, the  algebraic  equation  at  p.  530  will  here  become 

m'  +  A?n"  +  Bw,  +  C  =  0, 

which,  we  shall  suppose,  has  a  pair  of  equal  roots  represented 
by  mj  =  r/?2 ;  then,  shall 

y  =  Ci<?'"i^,     y-=z  a,^^"^^^,     or    y  =  Cie"'i^'  +  G^xe'^^'^j 
each  satisfy  the  preceding  equation  of  the  third  order. 


SECOND   AND   HIGHER   ORDERS.  533 

To  show  what  is  here  said,  it  will  evidently  be  sufficient 
to  show  that  the  differential  equation  is  satisfied  by  putting 
y  =  C.2a?<3"*i*,  which  gives 


^ 
(M 


and  ^  C2?7i/ir6'^*  ==  +  ZQ^ra^t 


Hence,  from  the  substitution  of  these  values  in 
we  get 


g-^S^^I+«^=«. 


Q.^e'^^^  i^m^  +  2Ami  4-  B)  ==  0  ; 

consequently,  because  m.x  is  one  of  the  equal  roots  of  the 
algebraic  equation  at  p.  532,  we  have 

m^  +  Km{-  +  Bmi  +  C  =  0, 
and  because  Swi*  +  2  Ami  +  B 

is  the  first  derived  or  limiting  equation  of  this,  one  of  its 
roots  must  also  equal  int}-^  and  thence  we  also  have 

Zm^  4-  2A??ii  H-  B  =  0,     and  thence    y  —  Q^j^e'^^'' 

satisfies  the  equation  as  it  ought  to  do.     (See  any  of  the 
treatises  on  Algebra,  on  the  equal  roots  of  equations.) 
It  is  hence  easy  to  perceive  that 

y  —  Cie"'i^  +  CaCCd'^i^  +  C36"*3*, 
must  satisfy  the  equation 

since  each  of  its  terms  satisfies  it ;  and  because  it  contains 
three  arbitrary  constants  it  must  be  the  complete  integral  of 
the  equation. 


534  DIFFERENTIAL   EQUATIONS   OF   THE 

Because  a  reasoning  similar  to  the  above  is  applicable  to 
each  term  of  y  that  results  from  any  number  of  equal  roots 
in  the  equation  at  p.  530,  it  follows  that  the  terms  of  y  re- 
sulting from  any  number  of  equal  roots  will  be  represented 
accordmg  to  the  directions  given  above. 

If  the  equation  at  p.  530  has  four  imaginary  roots,  such, 
that  two  corresponding  roots  of  each  form,  as  a  -\-h  \'  —\ 
and  a  —  h^—l^  are  equal;  then,  according  to  what  has 
been  shown  at  p.  531,  and  to  what  has  been  shown  above, 
they  may  be  expressed  by  the  forms 

e°*  [(Ci  +  Q>^)  cos  Ix  +  (Cs  +  C4a?)  sin  lx\  ; 

and  it  is  clear  that  we  may  proceed  in  much  the  same  way, 
when  the  equation  contains  any  number  of  pairs  of  equal 
imaginary  roots. 

Hence,  having  found 

y  =  Ce""^'  +  0,6"'='^  +  Ca^'^^a^  + 4-  C"^^n^ 

=  C,2/i  +  C,y,  +  03^3  + +  C'^y", 

for  the  complete  integral  of 
dry    .     .   d^'-^y       ^  d^-hi  ,^  dy 

whether  A,  B,  &c.,  are  functions  of  x  or  not,  yx^y.^^  &,o.,  being 
called  particular  values  of  y,  since  each  of  them  is  supposed 
to  satisfy  the  above  equations ;  then,  we  may  assume 

y  =  Ciyi  +  C,y,  + +  C"y 

for  the  complete  integral  of 

in  which  X  is  supposed  to  be  a  function  of  a? ;  by  supposing 


SECOND   AND   HIGHER   ORDERS.  535 

the  arbitrary  constants  Ci,  C^,  C3,  and  so  on,  to  vary,  by 
subjecting  them  to  the  following  conditions,  which  we  will 
illustrate  by  the  case  of  71  =  3. 

Thus,  the  equation  to  be  integrated  is 

whose  integral  we  suppose  to  be  represented  by 

y  =  Ci?/i  +  02^2  -f  Caya, 
supposed  to  be  subjected  to  the  following  conditions : — 
1st.  "We  have 

dx  ^  dx  ^  dx  dx  ^ 

by  assuming    yidCi  +  y^^^O.^  +  y^dCs  =  0. 
2d.  We  have 

d^'  -^''M  ^  ^'  dx^  "^  ^'  1^' 
by  assuming  the  equation 

dCidi/i  +  dC^dy.2  +  dC-^di/i  =  0. 
3d.  We  have 

d'y  _       d%  d^y.,  dy^ 

d^'-^''d^  ^^''d^'^  ^'  "(kf  ~  ^'        . 
by  assuming 

dC^d/y^       dG,d%       dCsd%       ^  _  ^ 
dx"     "^      dx'      "^      dx'      +  -^  -  ^• 

Hence,  if  we  determine  the  constants  from  these  three 
conditions,  we  shall  have 

y  =  Ciyi  +  Co2/2  +  C32/3 
for  the  complete  integral,  since  it  will  (generally)  contain  three 
constants,  as  it  ought  to  do.     It  is  manifest  that  we  may 


536  DIFFERENTIAL  EQUATIONS  OF  THE 

proceed  in  the  same  way  to  find  the  integral,  when  the  pro- 
posed differential  equation  consists  of  any  number  of  terms, 
or  is  of  any  order. 

Kemarks. — 1.  For  the  preceding  beautiful  method  of 
finding  the  integral  of 

^  +  A^  +  5^1-:^  + +m|^  +  %  +  x  =  o 

duf"  cbf-^       ^o?.zj"-2  dx         ^ 

from  that  of  the  same  equation,  when  X  is  omitted,  by  means 
of  the  variations  of  its  arbitrary  constants^  we  must  refer 
the  reader  to  p.  323,  vol.  i.,  of  the  "Mecanique  Analytique" 
of  Lagrange,  the  inventor  of  the  process  ;  reference  may  also 
be  made  to  p.  326,  vol.  ii.,  of  Lacroix ;  to  Young,  and  to 
most  of  the  late  writers  on  physical  astronomy. 

2.  On  account  of  its  importance  in  determining  the  lunar 
motions,  we  propose  to  give  a  different  and  very  simple 
method  of  finding  the  integral  of 

g  +  ,«=«  +  P  =  0, 

which  is  an  equation  of  the  second  order,  in  which  P  may 
involve  v  or  be  independent  of  it,  according  to  the  nature  of 
the  case. 

By  putting 
sin  mv  =  s     and     cos  mv  =  s\     we  have     6-^  -f  s'^  =z  1, 
which  gives  sds  +  s'ds'  =  0  ; 

and  by  taking  the  second  differentials  of  the  equations 

sin  mv  =  s    and    cos  rnv  =  s\ 
"we  ^Iso  have  the  equations 

■=-7;  +  ms  =  0    and    -^^-o  +  ^  *  =  ^* 
dv^  dv^ 


+  m^s'u  -\-  I  Yds'  =  mB  ■=  const. 


SECOND  AND  HIGHER  ORDERS.  637 

Multiplying  the  proposed  equation  bj  ds^  and  the  first  of 
these  by  cZw,  and  adding  the  products,  we  have 

whose  integral  gives 

dsdu  „  /*_, ,  . 

— j-g-  -j-  nv8U  +   /  rds  =  mA.  =  const ; 

and  in  like  manner,  from  the  proposed  and  the  second  of  the 
preceding  equations,  we  have 

ds^du 

Multiplying  the  first  of  these  by  5,  and  the  second  by  5',  and 
adding  the  products,  since 

sds  +  s'ds'  =  0     and    s^  +  s''  =  1, 
we  get    m^u  +  sjYds  +  s'jYds'  =  m  {As  +  Bs'), 
or  mu  =  A  sin  mv  -f 

B  cos  mv  —  sin  7nv  I  P  cos  7nvdv  +  cos  mv  I  P  sin  mvdv, 
for  the  sought  integral.     If 

p  _      M  +^         _  1 

such  that  M  and  m  represent  the  masses  of  the  sun  and  a 
planet  revolving  round  each  other  at  the  distance  r,  C'  rep- 
resenting the  square  of  twice  the  area  they  describe  around 
each  other  in  the  unit  of  time,  and  v  the  angle  r  makes  with 
a  fixed  line,  then,  if  7n  =  1 ,  our  integral  becomes 

1a-  -r,  M  +  m 

u  =  -  =  A  sm  V  -\-  B  cos  v  +  - — 7^, — . 
r  (j- 

If  in  this  we  put 

23* 


63S  DIFFEREXTTAL  EQUATIONS  OF  THE 

C'  =  (M  +  7n)p^     Ap  =  e  cos  w,     and    B/;  =  e  sin  Wj 

"we  shall  pret  r  =  :i ——, r 

°  1  -\-  e  cos  (v  —  to) 

for  tlie  equation  of  the  curve  described  by  m  in  its  revolu- 
tion around  M,  regarded  as  being  at  rest,  which  is  clearly  an 
ellipse  when  e  is  less  than  unity.  (See  Whewell's  "  Dy- 
namics," p.  27.) 

(7.)  If  a  differential  equation  between  x  and  y  involves 
only  the  simple  powers  of  y  and  dy  in  separate  terms,  and 
has  other  terms  that  are  independent  of  y  and  dy,  which  do 
not  involve  fractional  or  negative  powers  of  a?,  then  the 
proposed  equation  may  be  greatly  simplified  by  differentiating 
it  successively  on  the  supposition  that  y  is  a  function  of  x 
regarded  as  the  independent  variable. 

For,  since  (according  to  what  is  shown  at  pp.  11  to  13) 
dx  is  constant,  the  terras  that  do  not  contain  y  and  dy  will 
disappear  from  the  equation  in  consequence  of  its  successive 
differentiations. 

It  is  hencQ  manifest,  that  if  we  integrate  the  first  of  the 
differential  equations  that  is  freed  from  the  preceding  terms,. 
we  shall  (often)  readily  find  the  integral  of  the  proposed 
equation.     Thus,  taking 

7 

ady  =  ydx  +  Cx'dx,     or  its  equivalent     -—-  —  y  =  Gz^ 

(from  p.  217  of  Simpson's  "Fluxions"),  by  taking  its  suc- 
cessive differential  coefficients,  regarding  x  as  being  the 
independent  variable,  or  dx  as  constant,  we  have 

""dx^  dx  -  -^"^^  ""d?  ~d?-  ^^'  ^""^  ""  dx^  dx^  -  ^' 
thence  to  find  the  integral  of  the  proposed  equation,  we  must 
find  y,  such  that  it  shall  satisfy 


SECOND  AND  HIGHER  ORDERS.  539 

If  e  represents  tlie  base  of  hyperbolic  logaritbins,  and  D  an 
arbitrary  constant,  then  y  =  Dd"*^,  in  which  m  is  constant, 
reduces  the  preceding  equation. to 

1  - 

which  gives    tn  =  -;    and  thence  y  =  De^ ,   is  the   first 
d 

integral  of  the  preceding  equation,  and  of  course  a  part  of 

the  integral  of  the  proposed  equation.. 

Since  «^_^  =  20 

dx"^       dx- 

is  clearly  the  first  direct  integral  of 

having  20  for  the  constant,  it  is  manifest  that  y  =  Be**, 
having  D  for  its  constant,  must  represent  what  is  called  an 
indirect  integral  of  the  same  equation. 

To  find  the  remaining  terms  of  y,  or  the  proposed  integral, 
since  it  is  clearly  to  be  regarded  as  the  integration  of  an 
equation  analogous  to 

d^y        dry  _  ^^ 

dx^       dx^  ~  "   ' 

which  being  of  the  third  order  of  differentials,  its  complete 
integral  must  contain  three  arbitrary  constants;  conse- 
quently, we  may  represent  the  sought  terms  by 

y  =  Ax'  +  Ba;  4-  C, 
in  which  A,  B,  C,  are  the  constants,  and  the  integral  is  evi- 
dently of  the  proper  form,  since  it  must  be  supposed  to  have 
vanished  from  the  equation 


540  DIFFERENTIAL  EQUATIONS  OF  THE 

in  consequence  of  the  successive  differentiations  of  ttie  pro- 
posed equation. 

To  find  A,  B,  C,  we  substitute  the  values  of  y  and  -~  for 

them  in  the  equation 


and  thence  get 

(A  +  c)  ar*  +  (B  -  2aA)  cc  +  C  -  aB  =  0, 

which  must  clearly  be  an  identical  equation ;  consequently, 
we  must  have 

A  +  c  =  0,     B  -  2a A  =  0,     C  -  tiB  =  0 ; 

which  give 

A  =  -  c,     B  =  2aA  =  -  2ac,     Q  =  aB  =  -  2o?c. 

Hence,  from  the  substitution  of  these  values,  we  have 

y  =  Ax-  +  Bx  +  C  =  —  c{x^  +  2ax  +  2a^) ; 
consequently,  adding  this  to  the  preceding  value  of  ?/,  we 

X 

have  y  =  De""  —  c  {x^  +  2ax  +  2a-) 

for  the  complete  integral  of  the  proposed  equation,  which  is 
the  same  as  found  by  Simpson. 

By  a  like  reasoning,  the  integral  of  the  differential  equa- 
tion ady  =  ydx  +  cx'^dx, 
will  be  found  to  be  expressed  by 

X 

y  =  De''  —  c 
[i»"+naaj"-i4-n(^-l)aV-H7i(7i-l)  (n-2)aV-5+  &c.], 
which  agrees  with  Simpson's  integral;   noticing,  that   the 


SECOND  AND   HIGHER   ORDERS.  541 

integral  will  consist  of  an  unlimited  number  of  terms,  when 
n  is  not  a  positive  whole  number. 

(8.)  The  integral  of  a  differential  equation  of  the  second 
order  between  x  and  y  is  often  readily  effected  by  inter- 
changing the  dependent  and  independent  variables,  or,  which 

is  the  same  (see  p.  36),  since  -—-  and  -y-  are  equivalent  to 


^(l)=- 

dyd?x 
d^ 

and    d 

\dy)- 

dxdJ^y 
dy^ 

by  writing 

-'-1?  ^-  ^'y 

or     — 

dxd^y 
dy 

for    d'n 

Thus,  by  taking 
dxdy  - 

-  xd?y  ■ 

-ad?y 

xdy"- 
h     ■ 

=  0 

(from  p.  183  of  Yince's  "Fluxions"),  and  writing^ ^4 — 

for  d-y  in  it,  we  get 

dx      '       dx  h 


-,    ,         xdyd-x       adiidfx       xdiP- 


or  its  equivalent 

0 

Because  c^a?^  +  xd^x  =:  d{xdx), 

and  that  dy  is  constant  or  invariable,  by  taking  the  integral 
of  the  preceding  equation,  after  dividing  by  dy,  we  get 

xdx       adx        ^  _  r\ 
dy         dy        2b  ~     "^ 

an  arbitrary  constant.     Since  this  equation  gives 

7  2hxdx  2abdx 


2ho  +  x^       21)c  +  ar*' 


542  DIFFERENTIAL  EQUATIONS  OF  THE 

by  taking  the  integrals  of  this,  and  using  G'  for  the  arbitrary 
constant,  we  have 

y  +  C'=log(2JC+ar^)*  +  a|/M  X  &rc(rd=l  and  tan -^L.) 

which  agrees  with  Yince's  integral  after  the  constant  C  is 
added  to  his  value  of  y. 

For  another  example,  we  will  take  the  equation 

X(Px  —  dydx  =  0, 

which   is   such  that  the   method  of  integration   does   not 
readily  present  itself  to  the  mind. 

By  substituting -—  for  d^x  in  this  equation,  it  becomes 

dxd^y  7    7        /^  7    d^y       dx 

r-^  X  —  dydx  =0     or     c?aj  V^  H =  0, 

dy  ^  dy^        X  ^ 

in  which  dx  is  constant  or  invariable.     Since  this  equation 
is  the  same  as 

by  integrating  and  using  C  for  the  arbitrary  constant,  we  have 

dx      ,  ^  dx 

—  -7-  +  log  x=  C     or     dy  = 


dy         °  ^       log  a?  —  C ' 

in  which  the  variables  are  separated,  and 

/dx 
kg^'^^^ 

indicates  the  integral.     Putting 

dx 

log  x  —  C  =  3,     we  get    —  =  d2     or    c?aj  =  xdz, 

X 

which,  since  log  x=  C  +  z  gives  x  =6^'^'  =  e^e%   and  re- 
duces the  preceding  integral  to 


SECOND   AND   HIGHER   ORDERS.  543 

d3 


which  (see  p.  51),  gives 

^  f*/d2      dz      zdz       z^dz        „    \ 


.( 


^°""  + 1  + 1-  S  + 1-  lis  +  i-  lAi  +  ^')  +  ^"' 


in  which  C^  is  the  arbitrary  constant.  If  ?/  =  0  when  z  =  \^ 
and 

we  have  C  =  —  a^*^,  and  thence 

which  will  clearly  give  all  the  values  of  y  that  correspond 
to  those  values  of  z  or  log  x  —  0  that  do  not  differ  greatly 
from  unity,  or  that  are  positive  and  not  very  small.  (See 
Lacroix,  p.  512,  vol.  iii.) 

(9.)  Sometimes  a  differential  equation  can  be  integrated 
more  easily  by  eliminating  an  arbitrary  constant  from  it,  by 
means  of  its  differential  equation,  particularly  when  it  con- 
tains two   variables,  as   x  and  y,    and   higher  powers   of 

-~-    ox    -X-     than  the  first  power. 
ax  dy  ^ 

Thus,  by  taking  the  differential  of 


^^y%  +  {x'-Af-B)^-xi,  =  0, 


we  have 


g(.A..|  H-  ^-A^-b)  -.  (Ag  -m)(.|-.)=0, 


544  DIFFERENTIAL  EQUATIONS  OF  THE 

in  which  x  is  taken  for  the  independent  variable.    From  the 
proposed  equation  we  get 

aj2  _  Ay'*  -  B  =  xy -^ , 

dx 

which  being  substituted  in  the  preceding  differential,  and  re- 

dip- 
jecting  the  useless  factor  A  -j-^  +  1,  gives  the  differential 

equation 

xy^y  +  dy  {xdy  —  ydx)  =  0,     or    -  d^y  —  dyd  -  =  0. 

The  first  integral  of  this  clearly  is 

dx  y^ 

and  the  integral  of  this  is 

y'  =  Cx'  +  C\ 

From  the  substitution  of  the  values  of  —  and  y^  in  the  pro- 
posed equation,  after  a  slight  reduction,  we  get 
ACC  +  C  =  -  BC,     and  thence    C  = 


AC  +  1' 
consequently,  from  the  substitution  of  this,  we  shall  get 

for  the  integral  of  the  proposed  equation. 

Eemark. — For  the  substance  of  what  has  here  been  done, 
we  shall  refer  to  p.  123,  &c.,  of  Monge's  "  Application  de 
r Analyse  a  la  Geometrie,"  and  to  Lacroix,  p.  370,  vol.  ii. 

(10.)  The  integral  of  a  differential,  or  differential  equation, 
between  x  and  y,  may  sometimes  be  found  by  assuming  an 
expression  for  the  integral,  or  for  the  relation  between  x  and 


SECOND  AlTD   HIGHER   ORDERS.  545 

y^  in  ■undetermined  coefficients  and  exponents  of  x  and  y  if 
required,  such,  that  by  putting  the  differential  of  the  as- 
sumed integral  equal  to  that  proposed,  tliey  may  be  made 
identical,  so  as  to  determine  the  indices  and  coefficients. 
Thus,  to  find  the  integral  of  the  differential 
adx  +  hxdx 
ex  +  x^ 
(given  by  Yince  at  p.  184  of  his  "  Fluxions  "),  it  is  evident 
that  it  may  be  represented  by  the  form 
A  log  {ox''  +  «?'■  +  ^), 
of  like  form  to  the  integral  assumed  by  Yince,  in  which  A 
and  r  are  to  be  found. 

By  taking  the  differential  of  the  assumed  integral,  we  have 
.        Tcx^  ~  hlx  -f-  (/*  +  1)  x'dx 
^  ^^  +  x^-^'  ' 

which  must  be  made  identical  to  the  proposed  differential 
which  serves  to  find  the  unknown  A  and  '/'.  If  we  put 
r  =  5  +  1,  and  multiply  the  numerator  and  denominator 
of  the  proposed  differential  by  x%  its  denominator  becomes 
identical  to. that  of  the  assumed  differential;  consequently, 
we  must  have  the  identical  equation 

{ax"  +  Ix^  ^^)  dx  —.  \_A:rcxf  +  A{r  +  1)  cc^  +  ^]  dx^ 
or,   rejecting  the   useless   factor  x^dx^  we   shall   have   the 
identical  equation 

a  +  hx  =  Atg  -\-  A{r  -\-  1)  a?, 
which,  by  the  method   of  undetermined  coefficients,  gives 

b       T  -\-  \ 

a  =  Arc    and    h  =  A  (r  +  1\     and  thence    -  = , 

^  ^'  a  TO 

which  gives 

a              ,      .          a         1)0  —  a 
r  =  f and    A  =  —  = . 

OG  —    a  TO  G 


616  DIFFEREN"TIAL  EQUATIONS  OF  THE 

From  the  substitution  of  the  values  of  r  and  A  in  the 

assumed  integral,  and  using log  -^  to  stand  for  the 

arbitrary  constant,  we  get 


he 


I 


adx  +  l)X(lx       he  —  a  ^       ca?*<'-«  +  a/'' 
—  log 


ex  -\-  a^  0  Q 

for  the  correct  integral.     Putting 

a  bo 

in  which  x'  represents  some  particular  value  of  a?,  then 

be     1  6c  —  a 


/adx  +  Ixdx         ,        Icx^"- 


6c- 
6c 


l^^/6c-a    ^    aj^6c-.j 

which  equals  naught  when  x  =  x\  and  agrees  with  Vince's 
integral  when  it  is  properly  corrected. 

(11.)  Sometimes  the  integral  of  a  differential  equation  of 
the  higher  orders,  between  x  and  y,  may  be  simplified  in 
form  by  taking  a  function  of  the  independent  variable  for  a 
new  independent  variable.     Thus,  if  we  take 

dx'^   X  dx       ^^2/-^, 
and  put  1^  r=  a?"  +  2  =  a?*™, 

we  may  evidently  take  w  for  the  independent  variable. 

Because  y  is  supposed  to  be  a  function  of  a?,  which  equals 
a  function  of  u^  by  taking  the  differentials  of  these  equal 
functions,  we  have 

^dx=.'kdu      or     ^^^/i!f. 
dx  du      '  dx       du    dx^ 


SECOND  AND   HIGHER  ORDERS.  547 


consequently,  since  u  —  x""-  gives 

ax 

we  shall  ffet  ^-^  -—  ?n~  x^'-K 

°  ax  dti 


Again,  since  u  is  to  be  taken  for  the  independent  variable, 

in 


we  must,  for  —  in  the  proposed  equation  (see  p.  36),  write 

CL'X" 


d 

its  equivalent      J"'^— ;  consequently,  since 


dx 


dy  dy    ^ 

~  =z  m  ^  x' 
ax  du 


d'u 
and  that  v^  is  a  function  of  x. 
du 

^^      -m(;7i       \)  ^^x         +^^..yj 

ai^  air 

Hence,  from  the  substitutions  of  these  values  of 

di'^ 
dy  \dxj  ' 

dx''        dx 
in  the  proposed  equation,  after  an  obvious  reduction,  it  will 

become      -^^  +  (1  +  ' — )  -f 5-  =  0; 

which,  by  putting     1  -\ =  g    and    —^  =  h.     becomes 

du^  xidu  u  ' 


548  DIFFERENTIAL  EQUATIONS  OF  THE 

which  is  the  sought  transformation,  and  it  is  clearly  of  a 
much  simpler  form  than  the  proposed  equation. 
For  another  example  we  will  take  the  equation 

and  shall  take  u  =  e""'  for  the  independent  variable. 

T,  du       dy  du  ,  dy  dy   „, 

From     -/  =  -/  ^    we  have    -f  =n^  e"* 

ax       du  ax  ax  du      ^ 


Jdy 


\dx)  _    ^_^ 

dx      ~       du^  "        '    '"   du 


and  thence     — -. =  n^  Vi.  ^^"^  +  ^' 


^y.2nx    .    .2^2/ 


consequently,  from  the   substitutions  of  these  values,   the 
proposed  equation  becomes 


du"  "^ 
which,  by  putting 


du^       \  n)  udu        n^  u         ' 


1  H =  c     and     -^  =:  h 

becomes  -x-^  +  c  — r h-=zO, 

dw  udu  u 

which  is  the  same  as  the  transformation  in  the  preceding 
example;  consequently,  the  solution  of  one  of  the  given 
equations  must  be  given  by  that  of  the  other. 

Eemark. — The  preceding  equations  are  due  to  Professor 
Peirce ;  they  appear  to  have  been  first  published  at  p.  399 
of  Professor  Gill's  "Mathematical  Miscellany." 

(12.)  When  a  differential  equation  between  x  and  y  is 
such,  that  its  integral,  in  finite  terms,  can  not  easily  be  found, 
then  we  express  the  dependent  variable  in  a  series  of  terms 
of  the  independent  variable,  having  undetermined  exponents 


SECOND   AND   HIGHER   ORDERS.  649 

and  coefficients ;  and  then,  substituting  the  assumed  series  for 
the  dependent  variable  in  the  differential  equation,  we  deter- 
mine the  exponents  and  coeffieients,  so  that  the  indices  of 
the  independent  variable  shall  increase  from  left  to  right 
for  an  ascending  series,  and  shall  decrease  from  left  to  right 
for  a  descending  series. 
Thus,  to  integrate 

dv^  udu  u         ' 

the  transformation  found  in  (11),  it  is  easy  to  perceive  that 
"we  may  assume 

y  =  Au''  +  Bi^«  + 1  +  C2^«  +  2  +,  &c., 

which,  being  put  for  y  and  its  differentials  for  those  of  d^y 
and  dy  in  the  equation,  gives 

A\a{a-\-G—V)]u''-^-\-  [B  (a  +  1)  {a  +  c)  —  AA]  i^«  -  ^  + 

[C  (a  +  2)  (a  +  c  +  1)  -  BA]  u^  + 

[])(«  +  3)  (a  +  c  +  2)  -  CA]  t^«  +^  +  &c.  =:  0, 

which  must  clearly  be  an  identical  equation,  or  be  satisfied 
independently  of  u.  The  first  term  of  the  equation  is  evi- 
dently reduced  to  naught  by  putting 

«  =  0,     or     a  H-  c  —  1  ==  0, 

which  gives  a  =  1  —  <?,  and  A  is  arbitrary.  It  is  also 
manifest  that  the  remaining  terms  of  the  equation  will  be 
reduced  to  naught  by  the  equations 


D 


{a  +  l){a  +  Gy  {a+2){a  +  G+  1)' 

CA 


{a  +  3){a  +  G  +  2)' 
and  so  on.     If  in  these  expressions  we  put  a  =  0,  they  will 


550        DIFFERENTIAL  KQUATIONS  OF  TUE 

^  _  AA       -,  _         BA         _         AA» 
give    a -J--,    ^-i.2(c  +  l)-r:2o17Ti)' 


1.2.3c  (c  ■{-  l){o  +  2)' 

and  so  on  ;  and  in  a  similar  way,  by  putting  a  =1  —  c^ 
and  using  A'  for  the  corresponding  value  of  A,  the  same 
expressions  will  give 


D'  = 


1.(2-6')'  1.2.(2  -  c)  (3 -c)' 

A'A' 


1.2.3.(2  -  c)(3  -  c)(4  -  c)' 
and  so  on ;  which,  by  putting  2  —  c  =  G\  become 


D'  = 


1  .&'  1.2.  c'{g'  +  1)' 


1.2.3.6-'(c'  +  l)(c'  +  2)' 

&c.,  which  represent  the  values  of  B,  C,  &c.,  that  correspond 
to  A'. 

Hence,  according  to  principles  heretofore  given,  from 
the  substitution  of  the  preceding  particular  values  in  the 
assumed  value  of  y,  we  shall  get  the  complete  value  of  y 
expressed  by 

^=^^  P+  IT.  +  \IM^X)  +  1.2.3..(.  +  l)(c  +  2)  +^^-l  + 

^^      P+  Yd  +  L2.CV+1)  ^  1.2.3..V  +  l)(^-  +  2)^%^' 

in  which  A  and  A'  represent  the  two  arbitrary  constants, 
which  the  complete  integration  of  the  proposed  differential 
equation  requires. 


SECOND   AND   HIGHER   ORDERS.  551 

If  in  this  integral  we  put  tlie  values  of  u^  c,  and  A,  that 
correspond  to  the  equations 

'^l  +  ^^i^_BVy=:0,    and    f^  +  A^  -  BV"y  =  0, 

separately,  it  is  clear,  from  what  is  shown  in  (11),  that  we 
shall  get  their  integrals.  Hence,  if  in  the  first  of  these  in- 
tegrals we  put  A  =  0,  or  c  ==  1  —  - ,  it  will  be  the  integral 

of  the  equation  y^  —  Wx^y  =  0 ; 

and  by  putting  A  nz  0,  or  c  =  1,  the  second  of  the  pre- 
ceding integrals  will  be  the  integral  of 

It  may  be  added,  that  if  we  put  y  =  eP^^  {e  being  the 
hyperbolic  base),  we  shall  have 

^1  =  teP^-^     and  thence     ^  =  (c^^  +  Mx)  eP^^ ; 

consequently,  the  equation 

dhi 
g-BVy  =  0 

is  immediately  reducible  to 

dt  -f  tHx  —  B^a?"  =  0, 

which  agrees  with  the  equation  of  Eiccati.  (See  Lacroix, 
pp.  256  and  417,  vol.  ii. ;  and  Young,  p.  202.) 

From  y  =  e-/"^"^^  or  its  equivalent  log  y  =    I  tdx^  we  get 

dv 

t  =  —y- ;  consequently,  if  with 
ydx 


652  DIFFERENTIAL   EQUATIONS   OF  THE 

y  =  A  (1  +  i;^  +  &c.)  .+  AV-  (1  +  ^  +  &c.), 

the  integral  of  ^  —  B^x"y  =  0, 

after  a?"*  +  ^  has  been  put  for  u.  &c.,  we  proceed  to  get  — ~  , 

1/ax 

it  will  give  ^  in  a  function  of  x,  which  will  be  the  integral 

of  Riccati's  equation;   noticing,  that  although  A  and  A' 

enter  the  equation,  yet  its  form  is  such,  that  they  are  only 

equivalent  to  a  single  constant. 

(13.)  Sometimes  it  will  be  useful  to  determine  the  ex- 
ponents of  the  series  which  represents  the  dependent  varia- 
ble, particularly  when  the  series  is  descending ;  which  will 
readily  enable  us  to  determine  the  series. 

Thus,  to  find  the  integral  of 

By  assuming  x  =  Ay"  +,  &c., 

retaining  only  the  first  term,  we  have 

^  =  nAy--'    and    ^,  =  n  {n  -  1)  Ay—% 

and  thence  the  question  reduces  to 
2aAy^  -  n  {n  -  1)  aAy^  +  2Ay"  -  2/1^ Ay*  +  =  0, 

in  which  the  least  indices  of  y  are  clearly  ?i,  while  the  greatest 
indices  of  y  are  2n.  Putting  the  sum  of  the  coefficients  of 
the  least  power  of  y  =  naught,  we  have  2  —  71  (ti  —  1)  =  0, 
"whose  positive  root  gives  n  =  2,  which  enables  us  to  find  x 
m  a  series  of  ascending  powers  of  y.  For  2  put  for  n  in 
the  equation  reduces  it  to 

2a Ay-  -  2a Ay"  +  2 Ay  -  8Ay  =  0 ; 


SECOND   AND   HIGHER   ORDERS.  553 

consequently,  subtracting  tlie  least  indices  of  y  from  the 
greatest,  we  have  2  for  their  difference,  which  clearly  shows 

that  a?  =:  A^/'  +  ^if  +  C/  +,  &;c., 

is  the  proper  form  for  a?,  when  expressed  in  ascending  powers 
of  y.  For  a  descending  series,  we  put  the  sum  of  the  co- 
efficients of  the  highest  powers  of  n,  in  the  equation 

2aA2/"  —  n{n  —  l)  aAy'^  +  2A}y''''  —  2)vA?y-''  =  0, 

equal  to  naught,  and  thence  get  1  —  71^  =  0,  whose  positive 
square  root  is  ?i  =  1.     Putting  1  for  n  in  the  equation,  we 

have    2aAy  -  n  {n  -  1)  a  Ay  +  2Ay  -  2?i^  Ay  =  0 ; 

consequently,  subtracting  the  greatest  from  the  least  ex- 
ponents, we  have  —  1  for  their  difference,  which  evidently 

shows  that    x  =  A'y  -f  B'  +  O'y-^  +  D'y-'  +,  &c., 

is  the  proper  form  of  x  in  descending  powers  of  y.  (See 
p.  186,  &c.,  of  Vince's  "Fluxions.") 

Taking  the  first  and  second  differential  coefficients  of  the 
form  X  =  Ay^  +  By*  +,  &c., 

we  have        -7-  =  2 Ay  +  4By^  4-  6Cy^  +,  &c., 

and  |-^  =  2A  +  12By2  +  30Cy*  +,  &c. ; 

consequently,  substituting  the  series  for  a?,  and  these  values 
of  the  differential  coefficients  in  the  proposed  equation, 
after  properly  ordering  the  terms,  we  shall  have 

2^Ay2  +  2aBy*  +  2aCy^  +  2aDy^  +  ^ 

-  2aAy2  -  12t?By*  -  30aCy^  -  56aD/  - 
+  2Ay  +  4ABy«  +  (2B^^  +  4AC)  y^  + 

-  8 Ay-  32ABy«  -  {2>2W  +  (48AC)y«  - 
which  must  clearly  be  an  identical  equation. 

2i 


0, 


554  DIFFERENTIAL   EQUATIONS  OF  THE 

Hence,  we  must  have 


2aB  -  12aB  +  2A-  -  SA^  =  0,     or    B 


2aC  -  SOaC  +  4AB  -  32AB  =  0,     or     C 


2aA  —  2aA  =0,     or  A  is  arbitrary ; 

ba  ' 
3A« 

5a^' 
2aD-56aD  +  (2BH  4 AC)  -  (32B2+48AC)  =  0, 

31A^ 
^=-45^' 

and  so  on ;  consequently,   from  the   substitution   of  these 
values  of  A,  B,  C,  &c.,  we  shall  have 

for  X  expressed  in  a  series  of  ascending  powers  of  y^  in 
which  A  is  the. arbitrary  constant. 

To  find  a?  in  a  series  of  descending  powers  of  y^  we  have 

x  =  A'y  +  W  +  Q'y-^  +  J)'y-^  +,  &c., 

whose  differential  coejQ&cients  are 


dy 
d?x 


CV-^-2DV-3-,&c., 


and  ^  =  20^-'  +  6DV-''  +,&c; 

consequently,  from  the  substitution  of  these  values  in  the 
proposed  equation,  we  have 

2a A V  +  2aB'  +  2aCV-'  +  &c.  * 

~2aQ'y-^  —  ho.. 

2AfY  +  4A'BV  +  2B^2  ^  4,wC'y~^  +  &c. 

-  2A'y  +  4A'C'  +  4A'DV-'  +  &c. 

+  4A'C^  +  8A'DV-^  +  &c. 


=  a 


SECOND   AND   HIGHER   ORDERS.  555 

whicli  must  be  an  identical  equation.     Hence 

2A'2  —  2A'^  =  0,    or    A'  is  arbitrary ; 

2(^A' +  4A'B' =  0,    or    B'=       "^ 


2' 

2 


2aB'  +  2B'^  4-  8A^C'  =0,    or  .  C'  =  ^^ ; 

4B'C'  +  12A'D'  =  0,    or    D^  -  g^^, 

and  so  on ;  consequently,  from  the  substitution  of  tbese 
values  of  A',  B',  C\  &c.,  we  shall  have 

for  the  value  of  x,  when  expressed  in  a  series  of  descending 
powers  of  y,  in  which  A'  is  the  arbitrary  constant. 

Because  the  proposed  equation  is  of  the  second  order  of 
differentials,  its  complete  integral  must  involve  two  arbi- 
trary constants  ;  consequently,  from  the  addition  of  the  two 
particular  values  of  a?,  we  get  the  complete  value  of 

,    .,     3A=    ,      3A^   g       .  .,        a         a" 

0,  =  Ky-  ^t  +  g-^2/« -  &o.  +  A'y--  +  ^,-  +  ,&c., 

as  required. 

Eemarks. — 1.  If,  with  Mr.  Young,  at  p.  260  of  his  "In- 
tegral Calculus,"  we  integrate  the  equation  ll  +  -~jy  =  l 
by  the  preceding  methods,  we  shall  get 

y  =  l+^2{x-a)^-l{x-a)-\-^{x-af  -.  &c., 

O  ib 

in  which  h  is  the  value  of  y  that  corresponds  to  a?  =  a,  which 
is  clearly  equivalent  to  the  determination  of  the  arbitrary 
constant 


656  DIFFERENTIAL  EQUATIONS  OF  THE 

2.  This  question  can  clearly  be  integrated  without  using 
series,  bj  regarding  x  as  being  a  function  of  y.  For  the 
equation  can  be  reduced  to 

dy      ^  dx  y  ^1  .       —  1 

^  dx  ^'  dy        1—y  1— y  1— y 

which  gives  dx  =  —  dy  -^  - — — ; 

and  thence 

x  =  -y-log{l-2/)  =  -y  +  log  ^— — , 

which  needs  no  correction,  supposing  y  and  x  to  commence 
together. 

(14.)  To  what  has  been  done,  it  may  be  added,  that  differ- 
ential equations,  of  the  first  order  in  particular,  may  often 
be  elegantly  integrated  in  a  continued  fraction. 

Thus,  by  taking  the  differential  equation 

(See  Lacroix,  vol.  ii.,  p.  427),  and  putting 

A^      and    Ax^=::X,     P'=  P+ QX4-RX^  + S  ^, 

^       l-\-  y  dx 

Q'  =  2P  +  QX+S^,    E'  =  P,     S^=-SX, 
we  shall  have  the  transformed  equation 

p'  +  qy  +  ^Y-  +  s'%  =  0. 

3x^ 
If  in  this  equation  we  put  y'  =  - — '- — 77 ,   and  in  the  pre- 

ceding  results  change     P',  Q',  K',  S',    and     ~ , 


SECOND   AND   HIGHER   ORDERS.  657 

dij" 
into  Y\  Q",  E'',  S'',  -j~ ,   we  shall,  in  like  manner,  get 

for  the  transformation  of  the  preceding  transformed  equation, 
and  so  on,  to  any  required  extent. 

To  make  what  has  been  said  more  evident,  take  the  par- 
ticular example 

^y  +  (1  +  ^)  ^  -  0. 
Then  y  =  - — '- — ,,   supposing  A,7j^  and  y'   very  small, 
may  approximately  be  reduced  to  y  =  Aa?'',  which  gives 

and  thence  the  proposed  equation  is  approximately  reduced  to 
(mA  +  a  A)  x""  +  Aaa?«-^  =  0, 

which   is   approximately   satisfied  by  putting   a  =  0,    and 
omitting  mA  on  account  of  its  supposed  minuteness  ;  con- 

A 

sequently,  we  may  put  y  = j ,  and  shall  thence  get 

dy  _  dx 

dx~  ~  (1  +  yj ' 

Hence,  from  the  substitutions  of  these  values  of  y  and 

dy 

~- ,  the  proposed  equation  becomes 

which  is  easily  reduced  to 

dy' 
—  m  —  my'  +  (1  +  a?)  -^  =  0. 


658  DIFFERENTIAL  EQUATIONS  OF  THE 

By  changing,  as  before,  'i/  into  Ba?*,  and  -~-  into  JBa^~\ 
this  equation  becomes 

—  m  —  mBaj*  +  ^-Ba;*  +  JBoj*-^  =  0 ; 

"which  is  clearly  satisfied,  as  required,  by  putting  5  =  1,  and 
making  B  =  m,  when  terms  of  the  order  m-  are  omitted  ; 

consequently, 77  is  reduced  to ^ ;  noticing,  that 

>        if  ^        if 

we  have  hence  reduced  y  to 


y 


1  +  /  Baj«  ma? 


If  for  1/  in  the  equation 

-  m  -my'  ■\-{l  +  x)^=  0, 

we  put  its  equal,  after  a  slight  reduction,  we  shall  get  the 
equation 

{rn-l)x-\-\l  +  {m-l)  a^]  y"  +  y'"-^  {l+x)x  ^  =  0. 
Putting  Caf  and  cCaf  -  ^  for  y''  and  -|—  in  this,  we  get 

(m—  l)aj  +  [1  +  (m  —  l)a?]  Ca?<^  +  C'^ar'  +  cC{l  +  x)x' =  0; 

which,  by  putting  c  =  1,  omitting  the  common  factor  x,  and 
retaining  only  the  principal  terms,  reduces  to 

m  -  1  +  2C  =  0,     and     gives     0  =  -  '^-  ; 
consequently,  =——77,  becomes    ~^^^ — • 


Proceeding 

SECOKI 

in  this 
A 

)  AND   HIGHER  OB 

way,  we  shall  get 

LDEKS. 

^"1 

4-  rrix 

1- 

m  — 
1 

1    X 

'2 

1  + 

m  -{-  I   X 

3      '2 

^             771—2 

^            3 

X 

2 

1  +  "^ 

+  2  a? 

J-  -1 

5      '2 

1  - 

m  —  3 
5 

•2 

559 


1  +,  &c., 

for  the  sought  continued  fraction,  the  same  as  found  by 
Lacroix,  at  p.  429,  vol.  ii. 
Because  the  equation 

my  +  (l  +  a^)^  =  0     • 

,     ., .    ,  dy  mdx 

IS  reducible  to  ^j^  —  —  _ 

y  1  +  a?' 

.     .  ,     .  0 

Its  integral  gives  y  =  ^y—y^i', 

which  gives  y  =^  C  when  a?  =  0,  and  the  continued  fraction 
when  x  —  0  gives  y  =  A,  and  thence  C  =  A ;  consequent- 
ly, by  putting  A  —  1,  we  shall  have 

1^_  _  _J 

(1  +  aj)"*  ~"  1  +  7rix 


771  —  1    X 


1 ^ 

+,  &c., 


560  DIFFERENTIAL  EQUATIONS  OF  THE 

or,  taking  the  reciprocals  of  these  equals,  we  have 


(1  +  aj)"*  =  1  4- 


^       m  —  1  X 


1      '2 


m  -f  1  OS 


3      '2 

1 

m  —  2  X 
~      3      -2 

1+-+? 

a? 
•2 

i_» 

-3 
5 

X 

'2 

l+,&c. ; 

consequently,  the  binomial  theorem  may  be  considered  as 
being  reduced  to  the  form  of  a  continued  fraction. 

Since  the  exponential,  theorem  (5),  at  page  51,  reduces 
(1  +  xY  to 

-I    ■       1/1         N       m^  rioff  (l+a?)P        0 
1  +  m  log  (1  +  »)  + 12  "^'       ' 

we  hence  get 

1  +  m  log  (1  +  a^)  +  &c.  =  1  + ^__ 

■^      r~'2 


1  +,&c. ; 
or,  from  an  obvious  reduction,  we  have 

log  (1  +  £C)  +  &C.   =  - 


^       m  —  1   X 
1~*2 


1  H-,  &c., 


SECOND  AND  HIGHER   ORDEES.  5G1 

wliicli,»by  putting  m  —  0,  reduces  to 


log  (1+  aj)  r:r  . 


1  +  i^ 


^3   2 


^3  2 


^5  2 


^5  2 


1  +,&c. ; 


consequently,  the  hyperbolic  logarithm  of  1  +  a?  is  reduced 
to  a  continued  fraction. 

1-1 )  ,  when  w,  is  infinite,  equals 


[see  (J^)  at  p.  51],  it  clearly  follows  that  if  in 


(1  +  a^)-  =  1  4- 


inx 


^        in  —  \x 


1        2 

1  +,  &c., 

qn 

we  put  —  for  ,r,  and  suppose  w.  infinite,  we  shall  get 

24* 


662  DIFFERENTIAL  EQUATIONS  OF  THE 

X 


e'=:l  + 


1    X 

1*2 


^  ^  3   2 


3   2 


1  +  i.^ 
^5   2 


1   » 
5*2 


1  +,  &c., 

for  the  conversion  of  e'  into  a  continued  fraction. 
For  another  example,  we  will  find  the  integral  of 

1  -  (1  +  (B")  ^ ,     or  its  equivalent     dy  =  j-^ . 
Bj  taking  the  integral  of 

%  =  ^ — — ;;  =  ( 1  -—  :; I  d^!, 

,                            r    dx                     r  x^dx 
we  have         V  =    I  z, =  x  —    I , 

^       J  1  +  x^  J  l  +  af"' 

which  needs  no  correction,  supposing  x  and  y  to  commence 
together;   and  to  find  y  in  a  continued  fraction,  we  may 

clearly  put  y  =  ~- — ;-,  which  gives 


aj  r  x^'dx  1  ^        1    r  x^dx 

1  -\-  y'  •/l-raj"'  I  +  y'  x  J  1 -\-  x' 

whose  reciprocal  gives 

^  \         xJl  +  x"") 

=  14-1/*  ^^^  ^  A  _  1  r^^x\ 

xJl  +  x^'\         xJ  1  +  xY 


SECOND   AND   HIGHER   ORDERS.  563 

^^  ^  ~xJ  r+ic^  '  \  ~  xJ  r+~^/ 

""  U  + 1     X  J  iT^/  ~  I  ^  xJ  i  +  ^j ' 

A 

consequently,  by  putting  :^ -,  =  y\  we  thence  get 

aft 

and  thence  A  =  — — r  is  the  numerator  of  the  second  of 

n  +  1 

the  continued  fractions,  and 

or,  taking  its  reciprocal,  we  have 

.    ,     „       /,        \    r  x'^dx  \        r       n  +  1    r  x-^'dx  \ 

~     '^[x''-^J'l  +  X''      xJ  1  +  W   •  I  ~"  S^^^y  i"+a^7' 

or,  after  a  slight  reduction, 

„  ^  l{n-^l)x^ ^_  _  n+_l    r  x^Hx        1    r^dx\ 

^    ~  \  2/1  +  1  n  4-  1        a"  +  ^  y  1  +  a"  "^  a^y  1  -f  ic"j 

""  [{n  +  l)(2n  +  1)        c^y  1  +  ^"  "  "a^^^y  1"+^/ 
/         n  -\-l    r  x-'^dx  \ 

Putting  y''  =  — — JJ-, ,   we  shall  easily  get 
B  ^ 


(/I  +  1)  (2n  +  1) 


564  DIFFERENTIAL  EQUATIONS  OF  THE 

for  the  numerator  of  the  third  of  the  continued  fractions ; 
and  then  taking  the  reciprocal,  we  shall  have 

^^y  \         aj"  +  ^  J  l  +  x-l 

and  so  on,  to  any  required  extent.     Hence,  we  shall  have 
dx  X 


^=A 


+  aj"       1  .      ^" 


71+  1 

1     1 

tvx'' 

^   '    {n^ 

-  1)  (2?i  +  1) 

1  + 

{n  +  1)-V 

(2/1  +  1)  (3^1  +  1) 

(2^)V 

'   (371  +  1)  (471  +  1) 

1+,&C., 

for  the  sought  continued  fraction.      (See  Lacroix,  vol.  ii., 
p.  431.) 

If  71  =  1,   this  formula  gives   the  same  expression  for 
log  (1  +  x\  as  at  p.  660 ;  and  if  ti  =  2,  we  shall  have 

dx 


f 


3 \,   —    wux  it/  — r— 

"^1.3 


•+s 


16^ 
^  7.9 


1  +,  &a 


SECOJTD  AN'D   HIGHER  ORDERS.  565 

(15.)  We  will  now  proceed  to  show  liow  to  find  the  inte- 
grals of  what  are  called  Simultaneous  Equations^  such  as 

and  Wy  +  N'a^  +  P'  ^  +  Q'  ^  =  T', 

in  which  M,  N,  P,  &c.,  are  supposed  to  be  functions  of  ^, 
considered  as  being  the  independent  variable ;  the  equa- 
tions being  coexistent  and  of  analogous  forms. 

1.  To  integrate  this  kind  of  equations,  after  multiplying 
by  the  differential  of  the  independent  variable  dt^  they  may 
be  written  in  the  forms 

QLy  +  Na?)  dt  +  Ydy  +  (^Ix  =--  "Yldt, 

and  (M^  +  Wx)  dt  +  Y'dy  +  Q!dx  =  Tdt ; 

and  multiplying  the  second  of  these  by  0,  regarded  as  being 
a  function  of  t,  and  adding  the  product  to  the  first,  we  get 
the  single  equation 

[(M  +  M'^)  2/  +  (N  +  ^'0)  x]  dt  + 
(P  +  P'^)  dy  -f  (Q  +  Q'^)  c/^  =  (T  +  TB)  dt. 
Putting 
M  +  M^0  =  Ml,     N  +  N^6?  ==  Ni,    P  +  P^0  =  Pi, 
Q  +  Q'0  ^  Qi,     T  +  T'0  =  Ti, 
we  thence  have 

(Miy  +  Ni,^)  dt  ■\-  V^y  +  q,dx  =  T,dt, 

a  form  analogous  to  the  proposed  equations,  and  it  is  clear 
that  this  equation  is  equivalent  to 

Ml  {y  +  ^  x)  dt  +  Pi  (^dy  -\-^dx^  =  T,dt, 


666  DIFFERENTIAL  EQUATIONS  OF  THE 

N 
Putting  y  +  ^x=z 

and  assuming       diy  +   ^i— x\  =  dy  +  ~  dx^ 
the  preceding  equation  is  reduced  to  tlie  form 

'M.^zdt  +  PiC?2  =  T A     or     dz  +  ^  zdt  =  ^  dt, 
which  (see  p.  455)  is  a  linear  equation.     From 

by  taking  the  indicated  differentials,  we  have 

dy  +  ^^dx-{-  xd^^  =  dy+  ^^dx, 

or  =-^  dx  +  xjI  ^  =   ~  dx, 

Ml  Ml       Pi 

which  must  be  an  identical  equation ;  consequently, 

^  =  1'     and     cZ^'  =  0, 
Ml       Pi  Ml 

N  +  N'0       Q  +  Q'0  ,     ^^  +  We       . 

^^        MTM^  =  PTT^     "^^    ^MTM-^  =  ^' 

and  by  performing  the  indicated  differentiation  of  the  second 
of  these  equations,  and  eliminating  6  from  the  result  and  the 
preceding  equation,  we  shall  get  the  relation  between  the  coef- 
ficients of  the  proposed  equations  that  must  exist,  in  order 
that  their  integration  may  be  reduced  to  the  integration  of  the 
preceding  linear  differential  equation  of  the  first  order.  It 
may  be  noticed  in  this  place  that,  if  we  integrate  the  equation 

"^  MTM^""    '   we  shall  have   ^^^p^  =  C  =  a  constant, 


and  tlience 


SECOND  AND   HIGHER  ORDERS.  667 


reduces  to  F+"P'^  ~  ^ ' 

consequently,  eliminating  0  from  these  equations,  we  Lave 


,^N-CM     ^^^    ,    _Q-CP 


/> 


CM'-N'  CP'  +  Q 

,  .  ,     .  N  -  CM         Q  -  CP 

whicligive  ^^^^-^-^  =  ^^^-^^^ 

or,  reducing  this  to  a  common  denominator,  &c.,  we  have 

,   NP^  -  PN  +  MQ-  -  M^Q     _  KQ^-QN^ 
"^  PM'-MP'  PM'-MP^' 

Hence,  having  found  C  from  the  solution  of  this  quadratic, 

and  taken  the  differential  of  its  value  on  the  supposition  of 

the  constancy  of  C,  we  shall  clearly  get  the  same  result  as 

from  the  preceding  method. 

Solving  the  linear  equation  will  clearly  give  s  in  terms  of 

W 
t ;  and  thence  from  y  +  ^,  x  =  b,  we  can  find  y  in  terms 

of  X  and  t,  which,  being  substituted  in  either  of  the  proposed 
equations,  will  give  a  differential  equation  in  x  and  i,  whose 
integral  gives  x  in  t,  and  thence  having  found  x  and  y  in 
terms  of  t,  by  eliminating  t  we  shall  get  y  in  terms  of  x,  as 
required. 

2.  If  the  coefficients  M,  N,  P,  &c.,  in  the  first  members  of 
the  proposed  equations,  are  all  constant,  it  is  clear  that 

M  +  M' 

is  satisfied  by  supposing  that  6  is  constant ;  and  thence,  from 
the  solution  of  the  equation 


568  DIFFERENTIAL  EQUATIONS  OF  THE 

-we  shall  get,  by  the  solution  of  a  quadratic  equation,  two 
constant  values,  O'and^'',  of  0 ;  consequently,  if  m  and  n  are 
the  coefficients  of  the  linear  equation  that  correspond  to  0\ 
and  m'  and  n'  are  those  that  correspond  to  d'\  the  linear 
equation  gives  the  two  linear  equations 

dz  +  mzdt  =  ndt    and     dz  +  m'zdt  =  n'dt. 

Integrating  these  equations  by  the  formula  at  p.  455,  we 
shall  have 

z  =  e-/""^'(fne/'''^'dt\    and    z  =  e-f'^'^'(fn'ef'^'^'dt\ 

for  the  sought  integrals ;  noticing,  that  the  arbitrary  constants 
are  supposed   to  be   comprehended  by  the   integral   signs 

/  77,  &c.,   /  n\  kQ.     By  substituting  the  values  of  y  +  ^,  a? 

that  correspond  to  those  different  values  of  2,  for  z  in  the 
preceding  integrals,  we  shall  have  two  equations  in  a?,  ?/,  and 
^,  which  will  clearly  give  x  and  y  in  terms  of  t ;  consequently, 
from  the  elimination  of  t^  we  shall  get  y  in  terms  of  a?. 

3.  For  another  example,  we  will  integrate  the  simultaneous 
equations 

'     dy  +  (My  +  Naj  +  P^)  dt  =  Tdt, 

dx  +  {Wy  4-  N'a?  +  F'z)  dt  =  Tdi,  . 
dz  +  (M'V  +  N^'aj  +  F^'z)  dt  =  T'dt; 
which  may  be  supposed  to  be  obtained  from  three  equations 
of  forms  analogous  to  those  of  the  preceding  example,  by 

eliminating    -j-     and     j-     from  the  first  by  means  of  the 

second  and  third  equations,  and  so  on  for  the  remaining 
equations. 


SECOND   AND   HIGHER   ORDERS.  569 

Supposing  T,  T',  T'^,  to  be  functions  of  t^  while  the  other 
coefficients  in  the  preceding  equations  are  constant;  then, 
multiplying  the  second  and  third  equations  by  the  constants 
C  and  C,  and  adding  the  products  and  the  tirst  together,  we 
shall  have  a  single  equation  of  the  form 

dy  +  Qdx  +  G'dz  +  Q  (2/  +  Raj  +  S,2)  dt  ==  JJdt. 
If  in  this  we  change  C  and  C^  into  R  and  S,  it  will  become 

dy  +  'Rdx  +  Sc^3  +  Q  (?/  +  Ra?  +  Ss)  dt  =  XJdt ; 
consequently,  putting    y  +  Hx  +  Sa  =  'y, 
since  R  and  S  are  constants,  the  equation  will  become 

dv  +  Qvdt  =  JJdt, 
a  linear  equation,  whose  integral  gives  Vj  or  its  equal 

y  +  'Rx  +  Sb, 
equal  to  a  function  of  t. 

4c.  The  preceding  process    is    applicable  to   differential 
equations  of  the  higher  orders,  which  may  clearly  be  re- 
duced to  those  of  the  first  order. 
Thus,  the  equations 

d'y  +  (My  +  N;7?)  df  +  {Fdy  +  Qdx)  dt  =  Tdt"" 
and  d'x  4-  (M^  +  Wx)  df  +(:?'dy  +  Q'dx)  dt  =  Tdt', 
by  putting  dy  —  pdt    and     dx  =  qdt^ 

are  reduced  to  the  equations 

dy  ^  jpdt^     dx  =  qdt^ 
dp  +  Q^y  +  Naj  +  P^  +  Qq)  dt  =  Tdt, 
and         dq  +  Q^'y  +  Wx  +  P>  +  Q'^)  dt  =  Tdt, 
to  which  the  preceding  method  can  evidently  be  applied. 
(See  Young,  p.  264,  &c. ;  and  Lacroix,  p.  337,  &;c.) 


670  EQUATIONS  OF  THE  HIGHER  ORDERa 

By  reducing  the  first  two  of  the  preceding  equations  to 

dy  —  pdt  =  0,     dx  —  qdt  =  0, 

and  multiplying  the  second,  third,  and  fourth  by  the  con- 
stants C,  C,  C,  and   adding  the  products,  we  have  the 

single  equation 

dy-[-Odx-{-  C'djp  +C'W^  +Q(y+  Eaj  +  S/?  +Yq)dt  =Vdt. 
Putting 

dy-\-Qx  +  Q'p  +  0"q  =  d{y  +Ea;  H-  S/?  +  Yq\ 
and  C  =  E,     C  =  S,     C'^  =  V  ; 

then,  since  these  values  are  constant,  our  equation  is  reduced 
to  the  linear  form      dv  +  Qpjdt  =  JJdt^ 
in  which  v  is  put  for 

2/  +  Eaj  +  Si?  +  Yq, 
and  thence,  oy  taking  the  integral,  this  becomes  a  function 
of  t.  For  a  simpler  method  of  integrating  simultaneous 
equations  of  the  second  order,  under  certain  restrictions,  we 
shall  refer  to  p.  130  of  Whewell's  "  Dynamics,"  or  to  any 
other  work  that  treats  of  the  very  small  vibrations  of  what 
are  called  Complex  Pendulums, 


SECTION  IX. 

INTEGRATION    OF    DIFFERENTIAL    EQUATIONS    CONTAINING 
THREE   VARIABLES. 

(1.)  If  we  have     Fdx  +  Qdi/  -f  ^ds  =  0, 
such,  that  X  and  y  are  considered  as  independent  variables, 

P  Q 

and  z  a  function  of  them,  then,  if  i?  =  —  p  and  ^  ==  —  ^j^- , 

the  equation  will  be  reduced  to  the  form  dz  ^  pdx  +  qdy. 

If  this  is  the  total  differential  of  z,  regarded  as  being  a 

function  of  x  and  y,  it  is  evident  that  we  shall  have 

dz  ,  dz 

-J-    and     a  =  -J-. 
dx  -^       dy 

and  because  dz  is  supposed  to  be  an  exact  differential,  its 
equivalent  pdx  +  qdy  must  also  be  an  exact  differential ; 
consequently,  Euler's  condition  of  integrability  (see  pp.  489 
and  440)  must  exist,  which  gives  the  differential  coefl&cient 
of  p  taken  relatively  to  y  equal  to  the  differential  coefficient 
of  q  taken  relatively  to  a?,  and  thence,  since  p>  and  q  may 
contain  s,  we  shall  get 

dp_^dpd2^^dq^dj^dz     ^^    ^^^^^^_p^^ 
dy      dz  dy      dx       dz  dx  dy       "  dz       dx      -^  dz  ' 

or,  by  transposition,  we  have 


2?  =  -7-    and 


dy       dx       ^  dz       -^  dz 
for  the  condition  of  integrability  of  dz  =  pdx  +  qdy. 


572  DIFFERENTIAL  EQUATIONS 

Because  we  liave  supposed  that 


P=- 

P         , 

^  = 

Q 
B' 

we 

thence  get 

dy 

_      dy 

dy 

dq_ 

dx~ 

dx 
R^ 

c 

J 

't- 

Q 
■R 

dz 

dz 

^i- 

■R 

consequently,  from  the  substitution  of  -^ ,  -^ ,  &c.,  in  the 

ciy     ctx 

preceding  equation,  it  becomes 

dy  dy  dx  dx  dz  dz 

which  expresses  the  condition  of  integrabilitj  of  the  equa- 
tion Vdx  +  Q,dy  +  ^dz  =  0, 
on  the  supposition  that  when  multiplied  by  the  factor  ip ,  it 

P  Q 

is  reduced  to  dz  ■}-  ^  dx  -\-  -dy  =z  0, 

or  its  equivalent 

p  Q 

dz=^  —  ^dx  —  ^dy=z  j)dx  -\-  qdy^ 

which,  by  supposition,  is  an  exact  differential  equation. 
Hence,  to  find  the  integral  of  the  differential  equation 
Vdx  +  Q^y  +  Rf/s  =  0, 
we  examine  it  to  see  if  the  preceding  condition  of  integra- 
bility  is  satisfied ;  then,  if  it  is  satisfied,  we  multiply  it  by 
some  factor  m,  which  reduces  it  to  the  form 

mVdx  +  mQfly  +  inRdz  =  0. 


CONTAINING   THREE   VARIABLEa  573 

To  determine  tlie  proper  form  of  m,  we  may  omit  any 
one  of  tlie  terms  of  the  equation,  as  the  last,  then  we  find 

m  such  that  onPdx  +  mQdf/  =^  0 

shall  be  an  exact  differential,  on  the  supposition  of  the  con- 
stancy of  2 ;  and  putting 

du  =  mTdx  +  rnQdi/, 
by  taking  the  integral,  we  have 

u  =  J  {mVdx  +  mQ,di/)  +  0  (s)  =  V  +  </>  (2) ; 

in  which  0  (s)  =  a  function  of  2,  is  used  for  the  arbitrary 
constant,  since,  in  the  integration,  z  has  been  considered  as 
a  constant. 

To  find  (p  {z\  we  differentiate  the  members  of  the  equa- 
tion u  ~Y  -\-  <i>  (s), 
relatively  to  z  only,  and  thence  get 

die  _  dV       d4>  (z)  ^ 
dz  ~~  dz  dz     '' 

consequently,  since  u  is  here  supposed  to  be  the  integral  of 
mVdx  +  onQ^dy  +  rn^dz  =  0,     -y-  =  wE, 

O/Z 

and  thence 

^        dV    ,    d<t>{z)  d4>{z)  ^       dV 

mR  =  — -  4-  -^-^ ,     or     -^-^  =  mR , 

dz  dz  dz  dz 

which  gives       </>  iz)  =    I  ( wE  —  -j~j  dz, 

and  thence  the  integral  becomes  known. 

Because  0  (s)  is  independent  of  either  x  or  y,  it  is  clear 
that  wdien  the  factor  m.  is  correctly  found,  it  must  be  inde- 
pendent of  either  a?  or  7/. 


574  DIFFERENTIAL  EQUATIONS 

Thus,  to  integrate    ysdx  —  xzdy  +  yxdz  =  0. 
"We  have    P  =  ys,     Q  =  —  a?^,     and    K  =  ya?, 
and  thence  the  equation  of  condition  becomes 

dy  dy  dx  ax  az  dz 

yzx  —  yxz  —  yxz  -\-  xzy  —  xzy  +  yzx  =  0, 

and  the  condition  being  satisfied,  the   proposed   equation 
must  be  integrable. 

To  find  the  integral,  we  omit  the  last  term^  and  thence  get 
2  {ydx  —  xdy)  =  0, 

which  becomes  an  exact  integral,  by  multiplying  it  by  the 

1  zx 

factor  m  =  -"2 ,  the  integral  being  w  =  —  +  <^  (^)  J 

if  J 

^,  du       z       d(f)  (z) 

consequently,  _  =  _  +  -^J  ; 


or,  smce 


du  _yx  _x 


dz   -   -2     -    .,» 


X.  M      J.  XX       doiz) 

we  shall  get  -  =  -  4-  —^ , 

y      y         dz    ' 

which  gives  d<i>  {z)  =  0     and     <f)  (z)  =  const.  =  C, 

zx 
and  thence   u  = h  C   is  the  sought  integral. 

if 

For  another  example,  we  may  take  the  equation 

zydx  +  xzdy  +  xydz  +  azHz  —  0, 

which  will  be  found  to  satisfy  the  condition  of  integrability ; 
consequently,  its  integral  can  be  found.  Indeed,  since  the 
integral   of  the  first   three  terms  of  the   equation   is   xyz^ 

and  that  ~  is  the  integral  of  the  fourth  term,  it  is  clear 


CONTAINING  THEEE   VARIABLES.  575 

that  tlie  integral  of  the  equation  is 

w  =  a;2/2  +  y-  +  C  =  0, 

in  which  C  represents  the  arbitrary  constant. 

(2.)  If  the  equation  Ydx  +  Qc??/  +  Eg?s  =  0 
does  not  satisfy  the  condition  of  integrability,  then  it  is  clear 
that  one  of  the  variables  can  not  be  regarded  as  being  a 
function  of  the  other  two,  so  that  the  variables  can  not  rep- 
resent a  surface ;  yet,  as  shown  by  Monge,  they  may  repre- 
sent a  pair  of  integrals  which  depend  on  an  arbitrary  func- 
tion of  z  considered  as  being  the  dependent  variable. 

For,  as  shown  at  p.  513,  &c.,  by  regarding  the  dependent 
variable  z  as  being;  constant,  the  resulting  equation 

Vdx  +  (^dij  =z  0, 
admits  of  an  integral.     Hence,  multiplying 
Tdx  +  Qdy  +  Ec?2  =  0 

by  m,  as  before,  so  as  to  make  (Pdx  +  Qd^  m  =  0  em.  exact 
differential,  we  shall  have 

Tmdx  +  Qmdy  +  EmcZs  ■=  0. 
Putting  du  =  Fmdx  +  Qmdy, 

we  shall,  as  before,  get 

u  =    C(?mdx  +  Qmdy)  =  Y  +  </>  (2)  =  0 
for  one  of  the  integrals ;  and  taking  the  differential  coefficient 
of  this  relative  to  s,  we  get    -7 — | ^— - ,  which,  being  put 

equal  to  Em,  the  coefficient  of  dz,  in  the  equation 

Tmdx  +  Qmdy  +  E//?c?s, 

dY       dcp{z)       ^ 

gives  -7-  H ~  =  Em 

°  dz  dz 


676  DIFFERENTIAL   EQUATIONS 

for  the  other  equation ;  consequently 

Y+H^)  =  0    and     '^J.  +  ^i^)  = -Rm, 
^  ^  a  2  dz 

in  whicTi  ^  iz)  is  an  arbitrary  function  of  z^  which  satisfy  the 

equation  Vmdx  +  Qmdy  +  "Rmdz  =  0. 

Thus,  of  ydr/  +  zdx  —  dz  =  0^ 

regarding  z  as  invariable,  the  mnltiplier  m-  is  2,  and  thus  the 
equation  to  be  integrated  becomes 

2ydi/  +  2zdx  —  2dz  =  0; 

the  integral  of  its  first  two  terms,  regarding  z  as  const.,  is 

y'  +  2zx  +  0(2)  =  0, 


and  thence  -^  H — ~-^  =  B>m 

dz  dz 

P^-%     or     2.  +  ''#) 
«0  dz 


becomes    2«  +  ^P  -  -  2,     or    2«  +  "^-^  +  2  =  0, 


which,  by  putting   ^{z)  =  2^,   is  immediately  reduced  to 
a?  +  3  +  1  =  0,  the  equation  of  a  right  line. 
In  much  the  same  way 

2xzdx  4-  2yzdy  +  x^dz  =  0 
can  be  satisfied  by 

{0^  +  f)z  +  0(3)  ==  0     and     a^  +  f  +  "^^  z=  cc^ 

For  another  example,  we  will  take  the  equation 

xdx  +  ydy  dz 

x{x  —  a)  -\-  y  {y  ^^17)        z'  —  'g~' 


CONTAINING   THREE   VAEIABLES.  577 

Tliis  equation  can  be  immediately  satisfied  by  putting 
X  (x  —  a)  -\-  y  {y  —  h)  =  (p(2)  :=  a.  function  of  0, 
wbich  reduces  the  proposed  equation  to 

2xdx  +  2ydy  =  — ^^  c?s, 

whose  integral  is         x^  +  y^  =  2   I  — — -  dz. 

It  is  easy  to  perceive  that,  by  putting  0  (2)  =  —  (s  —  c)  0, 
the  integral  becomes 


x^  -\-  y^  =  -  s^  -{.  W,     or    a;2  +  j/f^  +  s'  =  B 


the  equation  of  a  spheric  surface,  in  which  W  is  used  for 
the  constant. 

It  may  be  added,  that,  if  we  put  y  =  x,  the  differential  is 

immediately  reduced  to =  ;r = ,     whose 

-^  z  —  c       2x  —  a  —  b' 

integral  is  clearly  3  —  0  =  C  {2x  —  a  —  h). 

(3.)  It  may  be  observed  that  the  differentials  and  their 
integrals  here  considered  being  of  algebraic  forms,  their  in- 
tegrals are  sometimes  called  algebraic  integrals. 

Algebraic  integrals  of  differential  equations  can  some- 
times be  obtained  from  the  simplest  principles. 

Thus,  to  find  the  algebraic  integral  of 


dxVl+i(^  -{-  dy  Vl  -{-f  =  0, 
or  of  its  equivalent 

we  may  proceed  as  follows. 

24 


578  DIFFERENTIAL  EQUATIONS 


By  assuming 
X  +1 


V    or    a^  —  {l  +  v)x-i-l  —  v=0, 


and  using  x  and  y  to  represent  its  roots,  we  shall,  from  the 
well-known  theory  of  equations,  get 

X  -{-  y  =  1  +  V     and    xy  =  1  —  v, 

whose  sum  gives         x  -{-  y  +  xy  =  2 

for  an  algebraic  integral  of  the  equation.     Because  x  and  y 

are  roots  of        ar^  —  (1  +  -y)  a?  +  1  —  -y  =  0, 

it  is  clear  that  we  shall  have 


/a^  _  a,  +  1  _    /y>  -y  +  1  _ 
^        x  +  1       -  ^        y  +  1       -"■' 

consequently,  by  erasing  this  common  factor  from  the  second 
form  of  the  proposed  equation,  we  shall  get  the  differential 

equation  {1  -{-  x)  dx  +  {I  -\-  y)  dy  =  Oj 

whose  integral  is 

X^   4-  y'^ 

a?  -f  y  H -^—  =  C  =  the  arbitrary  constant 

From  the   algebraic  equation  we  have  x  +  y  ==  2  —  xy^ 
which,  substituted  in 


-  +  y  +  ""-^  =  c, 


2-^  +  ^^'  =  C,     or    (^^  =  C-2, 


reduces  it  to 
2  —  xy 
or,  more  simply, 


{x  —  yY  =  Q'  =  constant^ 
an  integral  that  is  evidently  of  an  algebraic  form. 


CONTAINING  THREE  VARIABLES.         579 

Remark. — If  we  proceed  in  like  manner  to  integrate 


dx 


M=  =  o, 


we  shall  get     a;^  —  (1  +  -y)  a?  +  1  —  "^  =  0, 
and  thence  a?  +  .y  +  a?y  =  2 

for  the  algebraic  integral,  the  same  as  before.     Hence,  from 


i/    ^  +  1     ^  a/    y  +  ^ 

^    X'  -X  +1  ^    f  -y  ^V 

the  proposed  differential  equation  reduces  to 
dx     ^df^_^^^     ^^^^     {\  +  y)dx  +  {l-\-x)dy  =  0, 


1  +  a?       1  +  y 
whose  integral  is 

a;  +  2/  +  a?y=:C  =  the  arbitrary  constant, 

which,  by  putting  2  for  C,  becomes  x  -\-  y  -\-  xy  =i  2^  which 
is  the  same  as  the  preceding  algebraic  integral. 

For   another  example,  we  will   show  how   to  find  the 
algebraic  integrals  of 


dx  Vl  -V  ix^  -{■  dy  Vl  +  f  +  dz  \n^  +  z^  z=  0. 
Because 

cfo  vr+w=  dx  {X-  +  X)  /^^~ 

ty  putting  ^lt+i=^' 

we  have  a^  -{-  {I  —  v)  ay^  -\-  vx  —  v  =  0  ] 

and  supposing  x,  y,  and  2,  to  be  its  roots,  we  shall  have 

x  +  y-\-z  =  v  —  l,     xy  i-  xz  +  yz  =  V,     and    xyz  =  v ; 


680  DIFFERENTIAL  EQUATIONS 

consequently,  eliminating  v  from  these  equations,  we  shall 

have 

x}/  +  X3  +  y2  —  {x  +  y  +  z)  =  l    and    xi/  -\-  xz  -i-  yz  =z  xySj 

which  clearly  correspond  to  two  of  the  algebraic  equations. 
To  get  the  other  algebraic  equation,  we  reject  the  factor 


3   +    1  1 


ar»  -f  aj2  y"  +  y^  2'  -h  s^         )/v' 

which  is  common  to  all  the  terms  of  the  proposed  equation, 
and  thence  get  the  differential  equation 

(ar"  +  a;)  c?a?  +  (jr*  +  y)  dy  +  {z^  +  z)  dz  =  0; 
and  by  taking  the  integral  of  this,  we  have 

sc^       x^       1/       'i?       z^       z^  ,     , 

or       2  (ar^  +  f  -\-  z')  +  3  (a.-  +  2/'  +  ^\=  C, 
which  is  clearly  an  algebraic  equation,  as  required. 

For  the  last  example  of  this  method  of  finding  algebraic 
integrals,  we  will  take 


dx  Vi  -\-  x^  ^-  dy  ^1"+  y^  —  0. 

By  putting  a?'  =  x'   and  y'  =  y\   we  shall  change  the 
equation  to 

1  ^  /  4 A+'^      1  ^  ,  ./I  +  y"" 

Putting 

1  4-  a;'^       1  +  t/'^ 

y—  = ^  —  ^     ^e  get     a?'2  —  'ya?'  +  1  =  0, 

X  y 

whose  roots  being  x'  and  y\  we  have 

x'+y'  =  V,    or    ar^+x^^  =  v,    and     ai'i/'  =  1,    or     xY  =  1. 

Rejecting  the  factor  -  from  the  proposed  equation,  we  get 


I 


CONTAIN-INa  THREE   VARIABLES.  681 

dx'  +  dy'  —  0, 
whose  integral  gives  a?^  +  3/'  =  a;^  +  ?/^  ==  C ; 
and  thence  y?]!^  =1,     a?^  +  2/^  =  C, 

are  algebraic  integrals  of  the  proposed  equation. 

For  fuller  information  on  algebraic  integrals,  see  pages 
383-404  of  Professor  Gill's  "Mathematical  Miscellany," 
published  at  Flushing,  L.  I.,  during  1836,  1837,  &c. ;  and 
for  other  methods  of  finding  algebraic  integrals,  together 
with  their  applications  to  elliptic  functions,  see  the  "  Exer- 
cices  de  Calcul  Integral,"  of  Legendre,  and  p.  471,  &c.,  of 
Lacroix. 

(4.)  Eesuming    dz  =  -^dx  -^  -^dy  =  ^pdx  +  qdy, 

in  which  2  is  a  function  of  x  and  y,  considered  as  being  in- 
dependent variables,  so  that  dx  and  dy  (see  p.  34)  must  be 
constant  in  differentiating  the  equation ;  consequently,  by 
taking  the  differential  of  the  equation,  we  shall  have 

d^z  =z  —--  dx^  +  2  -^ — J-  dxdij  +  -^-^  dip- : 
dx^  dxdy        ^        dy^    -^  ' 

d^z         d?z  d''z  GlZ 

or,  representing    ^ ,     -^  =  ^--  ,     ;^„   by  r,  .,  and  t, 

it  becomes  d^z  =  rdx^  +  zsdxdy  -\-  tdy'^] 

and  from  —  =  p     and    -^  =  q 

,      ,  ,  dz       d^z  -J  d}z     -, 

we  also  have       a  -y-  =  -7-5  aa?  +  -j—r  ctV 
dx       dx^  dxdy 

.  ,  dz         d^z     -,         d'^z  y 

and  d-^  =   ,    ,    dx  -\-  -r-^  dy, 

dy        dydx  dy^    ^' 

or  their  equivalents 

dp  =  rdx  +  sdy    and     dq  —  sdx  +  tdy. 


582  EQUATIONS  OF  THREE  VARIABLES. 

If  we  differentiate  the  equation 

{x-  a)'  +  (y  -  hy  +  (2-  cf  =  R^ 
successively,  according  to  the  preceding  principles,  we  shall 
get    (x  —  a)  dx  -\-  iy  —  b)  dy  -\-  {z  —  c)  dz  =  0^ 
day"  +  dy^  -{-  dz^  +  (s  —  c)  d'z  =  0, 

and  Mzd^z  -\-  {z  —  c)  d^z  =  0, 

for  its  first,  second,  and  third  differentials;  and  so  on,  to 
any  required  extent  * 

It  is  evident  that  the  preceding  forms  will  be  very  useful 
in  finding,  by  reverse  processes,  the  second,  third,  &;c.,  inte- 
grals of  differential  equations  between  ai,  y,  and  2,  when  z 
is  a  function  of  x  and  y  regarded  as  being  independent 
variables. 


SECTION  X. 

PARTIAL  DIFFERENTIAL   EQUATION'S. 

(1.)  Integration  of  partial  differential  equations  of  tlie 
first  order  between  a?,  y,  and  2,  z  being  considered  as  being 
a  function  of  x  and  y,  regarded  as  being  independent 
variables. 

A  partial  differential  equation  between  a?,  y,  and  2,  is  said 

uz         nz 
to  be  of  the  first  order,  wlien  it  involves  -7-  or  -^ ,  or  both 

of  these  differential  coefficients,  together  with  constants  and 
one  or  more  of  the  variables,  according  to  the  nature  of  the 
case.  It  is  hence  clear  that  a  partial  differential  coefficient 
can  not  exist  between  only  two  variables,  as  x  and  y ;  since 
if  one  of  them,  as  y,  is  a  function  of  the  other,  the  coefficient 

-~  must  evidently  be  complete  or  total^  and  not  partial  or 

incomplete. 

(2.)  The  simplest  partial  differential  coefficient  that  can 
exist  between  ^,  and  a?,  y,  must  evidently  be  of  the  form 

—  =z  (2,  obtained  by  regarding  y  as  constant  in  the  differen- 

CLX 

tiation;  consequently,  reversing  the  process,  in  the  integra- 
tion, we  multiply  by  c?a?,  and  thence  get  dz  =  adx,  whose 
integral  gives  z  =  ax  -\-  <l)y  hj  using  an  arbitrary  function 
of  y  to  complete  the  integral;  since  y  was  regarded  as  con- 
stant in  obtaining  the  proposed  differential  coefficient. 


584  PARTIAL  DIFFERENTIAL   EQUATIONS. 

dz 
In  like  manner,  from  -^  =  Y,  a  function  of  t/,  we  get 

dz 
z  =  Yx  -f  (py ;  and  from  -y-  =  X  =  a  function  of  x,  we 

ccx 


have  25  =  /  X^  +  (f>y. 


fiXA.MPLES. 

dz 

1.  The  integi*al  of    -^  =  a^  +  yx  +  a   is  required. 

Multiplying  by  dx  and  integrating,  since  y  and  a;  are  in- 
dependent variables,  clearly  gives 

s  =  -g  +  y  ^  4-  «aj  +  «;^y. 

2.  To  integrate 

dz  _       2x  J     ^^  _  y 

The  answers  are 


s  =  log  (y"  +  aP)  -\-  (fyy    and    s  =  '/a?^  +  y^  +  «^a?. 
3.  To  integrate 

dz  1  ,     ^s  1 

and     -^ 


c2aj        |/(y3  -  aj2)  dy        ^{x^  +  y^)* 

The  answers  are 

X 

z  =  sin-^  -  +  </>y     and    2  =  log  [|/(ar'  +  y^)  +  y]  +  </>». 
4.  The  integrals  of 

^  =/(^j  2/)    ^^d    Z"  ~  ^  ^  ~  ^  function  of  a?  and  y 
are  required. 


PARTIAL  DIFFERENTIAL   EQUATIONS.  585 

The  answers  are 

z  =  J  f{x,  y)dx  -\-  <pi/    and     z  =  ±  J  ~  dx  -{-  (py. 

(3.)  We  now  propose  to  show  how  to  integrate  the  equation 

on  the  supposition  that  M  and  N  are  functions  of  a?  and  y. 
From  the  equation  we  readily  get 


ds  _  _M  /dz\ 

~     N  \dxr 

dz 


dy 


which,  being  substituted  for   x"   ^^ 

,         dz   ^         dz   ^ 

that  results  from  the  consideration  that  ^  is  a  function  of  the 
independent  variables  x  and  y,  gives 

,         dz  {  ,         M   ,  \       dz  '^dx  —  M^y 

dz  ^=  ~^-~  \dx  —  ^  dy\  ^=i ^. 

dx\  ^    ^l       djx  N 

From  what  is  shown  at  p.  513,  since  IS^dx  —  l£dy  is  a  dif- 
ferential between  x  and  y,  it  clearly  admits  of  a  factor  ?, 
which  makes  it  an  exact  diflferential,  denoted  by  du  ;  or,  more 

generally, ^^^^ ~ ,   being  a  differential  between  x  and 

2/,  admits  of  a  factor  m,  which  makes  it  an  exact  differential, 
denoted  by  dv ;  consequently,  we  shall  have 

I  (^dx  —  Mo?y)  =  c?w,     or    m  I — — ^— — )  =  dv. 

Hence,  eliminating 

mx-Udy,    or    N^^^<^, 

25* 


586  PARTIAL  DIFFERENTIAL  EQUATIONS. 

fix>m  the  value  of  dz^  it  will  be  reduced  to 

\   dz   ,  ,        ,         1  dz    , 

dz  =  -.TTz.  -J-  du.     or  to     dz  = j-  do. 

IN  dx  m  dx 

dz 
Hence,  since  -7-  is  arbitrary,  we  may  clearly  suppose  it  to  be 

so  taken  that  c?3  =  7^  -7-  du 

ZN  dx 


may  be  exactly  integrable,  and  of  course  ^ 

1   dz 

7;r7  -J-  =  Fw  =  a  function  of  w, 
IN  dx  ' 

and  thence  2  =  ^i^ ;  and,  in  like  manner,  from 

1  dz 
dz  ^=  —  -r-  dv 
m  dx 

we  shall  get  z  =  i})V,  a  function  of  v,  which  must  clearly  be 
the  same  as  the  preceding  function. 
Thus,  to  integrate 

.        '^(|)-2'(S)  =  ^' 

by  compaiing  it  to  the  general  formula  we  get 

M  =  —  y    and    N  =  a?, 
and  thence 

I  (Ndx  —  Mdy)  =  da    becomes     I  {xdx  +  yd/j/)  =  du^ 

which  gives        1  =  2     and     u  =  x^  +  y^] 

consequently,  z  =  <p  {a^  -{-  y"). 

Similarly,  from   7n  r  ^^  ~ ^)  =  dv, 

we  get  m(^^^'^-l)  =  dv, 

which  gives  m  =  2x  for  the  sought  factor,  and  thence 
v  =  a^  +  f', 


PARTIAL   DlFFEREi^TIAL   EQUATIONS.  587 

consequently,  we  thus  get  z  =  xp  {x^  +  3/-),  whicli  is  essen- 
tially the  same  as  tlie  preceding  result ;  and  from  what  is 
shown  at  p.  215,  2  ==  (/>  {x^  +  y^)  becomes  the  general  equa- 
tion of  surfaces  of  revolution,  when  the  axis  of  revolution 
coincides  with  the  axis  of  2. 

For  another  example,  we  will  take  the  equation 

dx       ^  dy~~    ' 
Comparing  this  to  the  general  equation,  we  get 

M  =  a?    and    N"  =  y,  ' 

which  reduce 

I  (Ndx  —  M.dy)  —  du    to    I  {^dx  —  xdy)  =  du ; 
and  putting  ^  =  — ^ ,  the  integral  becomes 

/ydx  —  xdy  _  ^  _ 

X 

and  thence  z  z=:  6  ^: 


also 

771 


/'Ndx  —  Mdy\     ,  /ydx  —  xdy\ 

I — ■ — ^;^^ -I     becomes    m  \^ -\  =  dv. 


which,  by  putting    m  =  - ,     gives     -  =  '?^, 

i/  J 

and  thence  s  =  (/>  -,  the  same  as  before.    (See  Young's  "Dif- 
ferential Calculus,"  p.  199,  &c.) 

(4.)  We  will  now  show  how  to  integrate  equations  of  the 

^-  _^©^<^(|)-^  =  «.     : 

on  the  supposition  that  the  variables  P,  Q,  E,  are  functions 


588  PARTIAL   DIFFERENTIAL  EQUATIONS. 

of  a?,  y,  z.     Dividing  the  equation  by  one  of  the  variables, 
as  by  P,  and  representing  the  quotients  p  and  p  by  M  and 

N,  it  becomes         ^+M^  +  N  =  0, 

or  its  equivalent  ^  -|-  M/^  +  N  =  0  ; 

and  from  dz  =  -^  dx  -\-  -^  dy 

we  also  have  dz  =  pdx  +  qdy ; 

consequently,  eliminating^,  we  shall  get 

dz  +  '^dx  =  q  {dy  —  Mc&»), 
in  which  $-,  being  clearly  arbitrary,  we  must  put 
dz  +  Nc(?aj  =  0    and    dy  —  M.dx  =  0. 
If  M  does  not  contain  z,  the  equation  dy  —  M.dx  =  0 
admits  of  a  factor  m,  which  makes  m  (<^y  —  Mdx)  =  0  an 
exact  differential,  whose  integral  gives 

F  (a?,  y)  =  C  =  constant. 
Hence,  if  N  does  not  contain  s,  by  eliminating  y  from 
F  (aj,  y)  =  C,  we  shall  get  y  in  a  form  that  may  be  ex- 
pressed by  y  —f  {x,  c),  which,  substituted  in  dz  4-  'Ndx  =  0, 

will  give  an  integral  of  the  form  z  =  —   I  Ydx,  V  being  a 

function  of  x  and  c ;   consequently,  the  indicated  integral 
can  be  found,  whose  constant  ought  clearly,  for  generality, 
to  be  an  arbitrary  function  of  the  constant  C. 
Thus,  to  find  the  integral  of 

dz  dz  ,/   «  o^ 

by  comparing  it  with  the  proposed  form,  we  have 
U  =  ^    and    N=-ai'-^±iQ. 

X  X  .  , 


PARTIAL  DIFFERENTIAL   EQUATION'S.  589 

Hence  the  equations 

ds  +  l^dx  =  0     and    dy  —  Mdx  =  0 
will  become 

dz  —  adx — ^     ^  =  0  and  dy— -  dx  =  0  or  ^^ — ^—  =  0 : 

X  X  X 

and  it  is  clear  that  the  factor  -  reduces  the  second  of  these 

X 

equations  to  — ^    ^  ^       =  d-  =  0, 

X  .  X 

whose  integral  gives    -  =  C     or    y  =  Gx. 

Consequently,  from  the  value  of  y  in  the  first  of  these 
equations,  we  shall  get 

dz=:adx^il  +  0"),      , 

whose  integral  may  clearly  be  expressed  by 

z=.ax  x/{l  +  C-)  +  0C, 

«^C  being  an  arbitrary  function  of  C. 

Because  C  =  -,  from  the  substitution  of  this  value,  we 
thence  have  z  =  a  Vx^  +  y^  +  0  - , 

X 

for  the  equation  between  a?,  y,  and  z.  Eesuming  the  equa- 
tion dz  +  ^dx  =  q  {dy  —  Mdx), 

on  the  supposition  that  the  first  member  does  not  contain  y, 
and  that  dy  —  Mdx  does  not  contain  s,  then,  if  we  have  the 
factors  7n'  and  m,  which  we  may  have,  such  that 

m^  {dz  +  l^dx)  =  du     and     m  {dy  —  Mdx)  =  dV 

shall  be  exact  differentials,  they  will  give 


590  PARTIAL  DIFF.ERENTTAL  EQUATIONS. 

dz  +  '^dx  =  —7     and     dy  —  Mob  =  —  , 
whicli  will  reduce  the  preceding  equation  to 

du  =^  Q  —  dV\ 

^  m 

consequently,  since  the  first  member  of  this  equation  is  an 

exact  differential,  the  second  member  must  also  be  an  exact 

differential,  which  it  may  be  (on  account  of  the  arbitrariness 

rn! 
of  g)  by  putting  q  —  equal  to  a  function  of  Y,  and  thence, 

'ffh 

by  taking  the  integral,  we  shall  have  u  =  (pY,  or  u  must  be 
an  arbitrary  function  of  Y. 
Thus,  if  we  take  the  equation 

dz       X  dz       ^  _  n 
dx      y  dy       a?  ~~    ' 

we  shall  have      M  =  -    and    N  = , 

y  » 

and  thence  the  equation 

dz  +  l^dx  =  gi  {dy  —  Mc?ic) 
will  become 

,        s  ,           / ,        a?  7  \        xdz  —  zdx         lydy  —  xdx\ 
dz--dx  =  q[dy--dx)  or =  q\^--^ ), 

and  thence  7n'  =  -  and  m  =  2y  give 

X 

J  -^ =  -  =  u    and     f^ydy  —  2xdx  =  7/"-  —  x" ; 

consequently,  we  shall  have     —  =z  <f>  (^^  ^  or)  for  the  sought 

X 

integral. 

It  may  be  added,  that  if  we  eliminate  q  from  the  equations 

jp  +  Mq  +  N.=  0     and     dz  =pdx  -f  qdy, 


PARTIAL  DIFFERENTIAL   EQUATIONS.  591 

we  shall  have      'Kdz  -f  No?y  =j9  (McZa?  —  dy) ; 

or,  since  jp  is  arbitrary,  as  before,  we  sball  get  the  equations 

^idz  +  '^dy  =  0     and     dy  —  Mdx  =  0, 

and  it  is  clear  that  we  may  proceed  with  these  equations  in 
much  the  same  way  as  before. 

Where  it  may  be  noticed  that  we  may  use  the  first  of 
these  equations  Kdz  +  'Ndy  =  0  instead  of  dz  +  l^dx  =  0 
(the  first  of  those  before  found),  since  the  second  equations 
are  identical.     If  we  take  the  equation 

dz      X  dz      xy       ^      .      .  ,^  x        .  '^      xy 

-y -  +  -^z=0,     It  gives    M  = and    N  =  -^, 

ax      a  dy      az  ^  a  az' 

and  these  reduce  the  above  equations  to 

2ydy  —  ^zdz  =  0     and     'iiady  +  2xdx  =  0, 

whose  integrals  are  y^  —  z^  and  2ay  +  x^ ;  and  thence  from 
ti,  =zz  (pY,  by  putting  y^  —  z^  for  u^  and  2ay  +  x^  for  V,  we 
shall  have  y^  —  z^  =  (p  {2ay  +  x^).  (See  p.  50  of  vol.  ii.  of 
"Wright's  "Commentary  on  Newton's  Principia.") 

(5.)  We  will  here  venture  some  remarks  on  the  integra- 
tion of  the  partial  differential  equations  of  the  second  order 
between  x,  y,  and  z,  when  z  is  considered  as  a  function  of  x 
and  y. 

1.  A  partial  differential  of  the  second  order  must  involve 
one  of  the  coefficients 

d'z      d^z       d^z  drz 


dx^''     dy'^''     dxdy       dydx'' 

at  the  least;  and  may  contain  other  terms  like  those  that 
are  contained  in  partial  differential  equations  of  the  first 
order. 


692  PARTIAL  DIFFERENTIAL  EQUATIONS. 

2.  The  method  of  integrating  equations  of  this  order  is,  in 
some  respects,  quite  analogous  to  that  of  integrating  partial 
differential  equations  of  the  first  order.  We  will  now  pro- 
ceed to  integrate  some  of  the  simpler  forms  of  equations  of 
the  second  order. 

8.  To  integrate  the  forms 

^-  —  0     —  —  0         d      ^^    —    ^^    _  0 

cb^  ~~    '     dy-  ~    '  dxdy  ~~  dydx  ~ 

The  first  of  these  equations,  multiplied  by  c?a?,  gives 

-—  =  0,     whose  inteojral  is    ^-  =  </>y  ; 
dx  ^  °  dx       ^  ^ 

which,  multiplied  by  dx^  gives 

dz  =  <^ydxj     whose  integral  is    2  =  <pyx  -{-  ipy  =  x(t>y  +  ypy ; 

in  which  <py  and  V^y  represent  arbitrary  functions  of  y,  which 
are  used  instead  of  the  arbitrary  constants  in  common  in- 
tegrations. 

By  proceeding  in  like  manner  to  integrate  the  second  of 
the  proposed  equations,  we  shall  have 

dz 

— -  =  (^a?     and     z  =  ycpx  +  V^j 

the  arbitrary  functions  being  here  functions  of  x. 

To  integrate  the  last  of  the  proposed  equations  under  the 
form 

=  0,     we  have     -j-  =  0,     and  thence     -j-  =  ^a?, 


dxdy         '  dx  ^  die 

whose  integral  gives  z  =  I  dx<px  +  ^y  ; 

d^z    ■ 
and  the  integrals  of  the  form  -j-j-  =  0,  are 


dz 
dy 


(fyy    and    ^  ~  J  dycpy  -\-  ipx. 


PARTIAL  DIFFERE2TTIAL   EQUATION-S.  593 

It  is  manifest,  that  in  this  way  we  shall  get 

I  =  fvd^  +  <t>y 

for  the  integral  of  -T-3  =  P, 

and  z  =    I   I  I  Fdx  +  0y  j  dx  +  tfyy 

is  thence  the  integral  of  ds,  and  we  have 

z  z=z    I  i(fix  -\-    I  Vdy\  dy  +  -^a? 

d^z 
for  the  second  integral,  resulting  from  -y-g  =  P ;  and  in  like 

manner  we  have 


f\4>y  +  y  P^^)  %  +  ^-^ 


for  the  integral  resulting  from    ,    ,    =  P ;  noticing,  that  the 
equations 

^_  d^'z       _  d^z       _  -p    o 


(P,  Q,  R,  &c.,  being  functions  of  x  and  y),  may  be  treated  in 
much  the  same  way. 

4.  -^  +  P  -7^  =:  Q,  in  which  P  and  Q  are  functions  of 

X  and  y,  can  also  be  easily  integrated. 

dz 
For  by  putting  —-  =z  u,  the  equation  becomes 
cix 

-T-  -}-  Pw  =  Q,     or    <^?^  +  Vitd^x  =  QdXj 
a  linear  equation,  whose  integral  is  expressed  by 
ic  =  6-/p^^  IfQef^^^'^dx  +  <Py); 


594  PARTIAL  DIFFERENTIAL  EQUATIONS. 

and  since  w  =  -7- ,  we  thence  readily  get 


z 


(See  page  455.)    It  may  be  added,  that  the  equation 

i!l_  +  P  ^  =  Q 
dxdy  dx         ' 

by  putting  -j-  =  w,  becomes 

- — I-  Pw  =  Q,     or    du  +  Vudy  =  Qc?y; 
ay 

whose  integral,  as  before,  is 

z  =    fudx  =    f\e-f^^^  ifQef^^Hy\  +  (px\  dx  +  i/»y. 

In  much  the  same  way  we  can  change 

+  P  ^-  =  Q     into     -J— J-  +  P  -T-  =  Q, 


smce 


dxdi/  dy  dydx  dy 

d^z  d^z 


dxdy       dydx 

dz 
(see  p.  22) ;  consequently,  putting  u  =z  -j-  we  shall  have  the 

du       _ 
equation    -3-  +  Pt^  =  Q,     or    du  -{-  Fudx  =  Qdx, 

and  thence 

z  =  J^udy  =    f\e-f^^^  (fQef^^'^dx)  +  <}>y]  dy  -\-  \px. 

5.  It  is  manifest,  from  the  elimination  of  f{ax  +  hy)  from 
X  =/{ax  H-  hy\  at  p.  26,  which  gives  the  equation 

a^  -  h—-  0 
dy  dx  ~    ^ 


PARTIAL   DIFFERENTIAL   EQUATIONS.  695 

an  equation  of  partial  differential  coefficients,  tliat  equations 
of  partial  differential  coefficients  of  the  first  order  must  re- 
sult from  the  elimination  of  arbitrary  functions  from  equa- 
tions, in  a  way  very  analogous  to  that  in  which  ordinary 
differential  equations  result  from  the  elimination  of  arbitrary 
constants  from  equations. 

Hence,  it  is  clear  that  in  finding  the  integrals  of  partial 
differential  equations,  we  ought  analogically  to  add  arbitrary 
functions  to  correct  the  integrals,  instead  of  using  arbitrary 
constants  for  that  purpose ;  noticing,  that  the  forms  of  the 
arbitrary  functions  must,  in  particular  cases,  be  determined 
from  the  nature  of  the  question. 

Thus,  if  we  take  the  partial  differential  equation 

adz       hdz  _ 
dx         dy  ' 

to  find  its  integral,  we  may  proceed  as  follows : — 

Eepresenting  -y^    and     -j^ ,  as  usual,  by  p  and  ^,  the  pro- 
ws? dy 

posed  equation  becomes  ap  -\-  l)q  ^=  1\    and  since  s  is  a 

function  of  x  and  y,  we  also  have 

^^  ^  ^  ^^  +  ^  %  =  I>d^  +  ^dy- 

Hence,  by  eliminating^  from  the  equations 

op  +  bq  ^=  1     and    pdx  +  qdy  =  dz, 

we  get  ■  q  {hdx  —  ady)  =z  dx  —  adz ; 

which,  on  account  of  the  arbitrariness  of  ^,  is  equivalent  to 
hdx  —  ady  =■  0     and  dx  —  adz  =  0  ; 

or  eliminating  dx  from  the  first  of  these  equations  by  means 

of  the  second,  we  have 

dy  —  hdz  =  0     and     dx  —  adz  ~  0, 


596  PARTIAL  DIFFERENTIAL   EQUATIONS. 

whose  integrals  will  clearly  be  of  the  form 

y  —  1)2  =  A    and    a?  —  a^  =  B ; 

consequently,  since  these  equations  must  clearly  be  coex- 
istent, we  must  have  A  =  </)B,  or  by  substituting  the  values 
of  A  and  B,  we  shall  have 

y  —  hz  =  (f>{x  —  as), 

for  the  sought  integral ;  indeed,  if  (as  at  p.  26)  we  eliminate 
the  arbitrary  function  denoted  by  0  from  it,  we  shall  get  the 

proposed  equation       a  -z — [-  h  -j-  =  1 

from  it ;  noticing,  that  from  what  is  done  at  p.  211, 

y  —  hz  =  (p  {x  —  az) 

is  plainly  the  general  form  of  the  equation  of  cylindrical 
surfaces,  in  which  the  nature  of  the  directrix  is  undeter- 
mined. If,  however,  the  equation  of  the  directrix  on  the 
plane  x,  y,  is  of  the  known  form  y  =fxj  then  the  nature 
of  the  function  </>  can  easily  be  found. 

For  by  putting  naught  for  z  in  y  —  bz  =^  (t>{x  —  az),  it 
becomes  y  =  (px;  consequently,  since  y  =.  fx  we  have 
<px  =fx^  which  gives  the  form  of  (^,  and  of  course  the  equa- 
tion y  —  hz  =  <t>{x  —  az)  will  become  of  the  known  form 

y  —  hz  —f{x  —  az). 

For  further  illustration,  we  will  show  how  to  find  the 

1    /.  dz  dz  ^ 

mtegral  of  —x  +  ^  =  px  +  q  =  0. 

Since  dz  =  pdx  +  qdy, 

by  substituting  q  =  —  ^J>a?,  from  the  preceding  equation,  for 
q  in  it,  we  shall  get    dz  =  p  {dx  —  xdy) ; 


PARTIAL    DIFFERENTIAL   EQUATIOIJTS.  597 

whicli,  on  account  of  tlie  arbitrariness  of  j?,  gives 
dz  =  0     and     xdy  —  dx  ^=  0; 

or,  multiplying  by  --, ,    '-1-^  =  0, 

By  taking  the  integrals  of  tbese  differentials,  we  have 

z  =  a    and     -  =  h : 

X 

consequently,  since  these  integrals  are  coexistent,  we  must 
have  a  =  (ph'j  or,  substituting  the  values  of  a  and  5,  we  shall 

have  3  z=z  (f)  -^  which,  if  we  please,  may  be  written  in  the 

X 

form  0-^2  =  - ;  which  belongs  to  what  are  called  conoidal 

X 

surfaces,  whose  right  directrix  coincides  with  the  axis  of  s, 
without  reference  to  the  nature  of  its  curvilinear  directrix. 
If  the  curvilinear  directrix  is  given,  together  with-  the 
position  of  the  axis  of  the  conoid,  then,  putting  z=:  u^  we 
shall,  from  the  equations  of  the  curve  of  double  curvature, 
which  represent  the  curvilinear  directrix  and  the  axis,  find 
»,  y,  and  2,  in  terms  of  u ;  consequently,  having  found  x 

and  y  in  terms  of  s,  we  shall  get  -  in  a  function  of  b,  and 

X 

shall  thence  get  0~^3  in  a  known  form,  which  will  give  <p  in 

z  =  (})-,  3LS  required. 

Again,  resuming  z  =  ax  -\-  (py,  from  p.  583,  which  is  the 

integral  of  the  partial  differential  equation  -—  =  a;  then,  it 

ccx 

is  plain  that  there  is  nothing  in  the  nature  of  the  question 

to  determine  the  form  of  the  function  denoted  by  ^  in  (py, 

so  that  by  putting  x  =  0  we  have  2  =  (py  for  the  equation 


698  PARTIAL   DIFFERENTIAL   EQUATIONS. 

of  the  section  of  the  proposed  curve  surface,  by  the  plane 
s,  y,  between  a?,  y,  and  2,  of  an  entirely  undetermined  form. 

It  is  also  clear  from  z  =  ax  +  (py,  that  the  surface,  when 
cut  by  planes  parallel  to  that  of  the  axes  of  x  and  2,  always 
gives  right  lines  which  are  parallel  to  each  other,  since  a  de- 
notes the  tangent  of  the  angle  which  each  of  the  lines  of 
section  makes  with  the  axis  of  x. 

6.  It  is  manifest  from  what  has  been  done,  that  the  inte- 
gral of  a  partial  differential  equation  of  the  second  order  be- 
tween a*,  2/,  and  s,  must  involve  two  arbitrary  functions, 
through  which  the  surface  represented  by  the  integral  must 
pass. 

Thus  (at  p.  592),  we  have  found  e  =  x<t>i/  +  ip7/  for  the  in- 

tegral   of  -^-^  =   0,   in  which  y  and  i/jy  are  the  arbitrary 

functions. 

JI  we  put  x  =  0,  the  equation  z  =  x(t>y  +  i/'y  becomes 
z  =  t/>y,  which  represents  the  section  of  the  curve  surface 
by  the  plane  of  the  -axes  2  and  y. 

Since  the  axes  of  the  co-ordinates  are  supposed  to  be  rect- 
angular, it  is  clear  that  (py  represents  the  tangent  of  the 
angle  which  the  line  of  common  section  of  the  surface  by  a 
plane  parallel  to  the  axes  of  x  and  z  makes  with  the  axis 
of  x. 

Hence,  if  a  line  is  drawn  in  the  plane  of  the  section 
through  the  point  where  it  cuts  the  curve  2  =  Vy^  supposed 
to  be  drawn,  at  will,  to  make  an  angle  with  the  axis  of  a?, 
having  (py  equal  to  its  (natural)  tangent,  the  line  thus  drawn 
will  represent  the  common  section  of  the  plane  and  the  sur- 
face whose  equation  is  ^  =  x(f)y  ^-  V'y,  and  thence  we  may 
readily  perceive  how  the  curve  surface  may  be  supposed  to 
be  described  geometrically. 


PARTIAL  DIFFERENTIAL   EQUATIONS.  599 

7.  It  may  be  added,  in  concluding  this  treatise,  that  the 
integral  of  a  differential  equation  containing  any  number  of 
variables,  whether  they  are  total  or  partial,  may  clearly  be 
found  by  Maclaurin  s  theorem,  as  explained  in  {h')  given  at 
p.  25. 

8.  Sometimes  the  generating  function  of  the  integral  thus- 
found  can  be  obtained,  and  thence  the  integral  will  be  ex- 
pressed in  finite  terms. 

Thus,  if  we  have 

„       dz  cPz     a?^         d^z       a?  . 

^  =  ^  +  ^*  +  ;pr:2  +  d?  rro  +'  '^'=- 

in  which  z  is  expressed  in  terms  of  x^  supposing  it  to  be  a  func- 
tion of  a?  and  y  regarded  as  being  independent  variables,  and 

dz   d?z     .  A  ^    x.   ^\         ^         p      dz  dfz    . 

z,  -T- ,  -T-s ,  &c.,  are  supposed  to  be  the  values  oi  z,  ~r  ,-r^,  &c., 
dx^ dx^  ^^  dx  dx^ 

when  X  is  put  equal  to  naught  in  them ;  noticing,  that  if 

dz     d^z 
x=.^  makes  any  of  the  quantities  2,  — ,  -y-^,  &c.,  infinite, 

(XX     dx 

then,  by  putting  x  -\-  a  for  x,  we  may  proceed,  as  before,  to 
find  the  expansion  according  to  the  ascending  powers  of  x. 
It  is  hence  clear  that  the  preceding  series  may  be  regarded 
as  an  integral  of  a  partial  differential  equation  between  x 
and  z,  in  which  x  and  y  (or,  indeed,  any  number  of  varia- 
bles) are  independent  variables,  and  ^  is  a  function  of  them, 
or  depends  on  them. 

If  the  preceding  series  has  been  obtained  from  a  partial 
differential  coefficient  of  the  first  order,  it  is  clear  that  z  will 
represent  an  arbitraiy  function  of  the  variables  supposed 
constant  in  the  differential  coefficient,  and  if  the  series  has 
been  obtained  from  a  differential  coefficient  of  the  second 

dz 
order,  it  is  plain  that  z  and  -^  will  each  be  arbitrary  func- 

cix 


600  PARTIAL   DIFFERENTIAL  EQUATIONS. 

tions  of  the  variables  supposed  to  be  constant  in  the  differ- 
ential coefficient,  and  so  on,  to  any  extent  that  may  be  re- 
quired. 

cPz  cPz 

9.  If  we  take  -j-^  =  <r  -7-^ ,  it  is  manifest  that  we  may 

find  2  in  a  series  after  the  following  manner. 

Thus,  by  taking  the  successive  differential  coefficients,  we 

have  —  =  <?-^  =  &  -^-  =  c?-^ 

da?  d'l/^dx  dxdif  e/y^ 

^^ 
and  <^*g  _  .    dH    _    ,      dx'  _    ,^ 

ob*  ~     dx^d'if' "~        dy^    ~~      dy*^  * 

d^2        ^d}z  ,        , 

Bince    -j-^  =  (f  -—      and  we  have 
cto'*  dy^ ' 

^^  ^,d3 

^  _    «     <f  ^2     _    ,      ^  _    4      ^^ 
d^~      'dMf  ~      ~Wf  ~  ^  ~d^  ' 

and  so  on,  to  any  extent. 

Hence,  using  0y  to  represent  the  value  of  z  that  corre- 

d^z 
spends  to  a;  =  0,  -7-2  corresponding  will  be  expressed  by  <^"y^ 

dz 
and  using  -^y  to  stand  for  the  corresponding  value  of  -^  ,  and 

so  on,  it  is  manifest  that,  from  the  substitution  of  the  pre- 
ceding values  in  the  series,  we  shall  get 

^  =  <A2/  +  V^y  f  +  <?<^"y  ~  +  cW  xf  3  +  <^'ry  Y^^^  + 

for  the  integral,  or  the  required  expansion. 


PARTIAL   DIFFERENTIAL   EQUATIONS.  601 

If  in  this  series  we  put  (py  -\-  ipy  for  </>?/,  and  c  {(p'y  —  V' V) 
for  V^y,  it  will  be  reduced  to 

=  <i>{y  +  ex)  +  i)  {y  —  cx\ 

wliicb  expresses  the  integral  of  the  proposed  partial  differ- 
ential equation  of  the  second  order,  which  is  the  well-known 
formula  for  vibrating  chords.  (See  Lacroix,  voL  iL,  p.  639, 
and  Monge,  "  Application  de  1' Analyse  a  la  Geometric," 
p.  415.) 

To  find  the  total  integral  of  z  we  may  put  s,  Z,  — ,  — • ,  &c., 

CLX     OjX 

^'"^'  (^)'  (<!) '  ©  "^-^ ;  !•  ^i'  5 '  *«■•  ^^^  ^°  °"' 

„     ldX\     /d'X\     /(PX\     ,  J  .    „„ 

^°'  VSy)  '    \d^y)  '    W)  '  *=••  '■''^  '"  ''"'  '"^  (*  >''*  P-  ^'' 

noticing,  that  we  may  put  x  +  a,  y  -\-  b,  kc,  for  x,  y,  &c., 
and  that  in  this  way  the  integral  with  reference  to  all  the 
independent  variables,  or  any  number  of  them,  can  be 
found  at  will 

2« 


APPENDIX. 

To  complete  tlie  work,  we  add  the  following  important 
articles : — 

I. 

To  what  are  oilen  called  the  singular  points  of  curves,  we 
add  the  following  from  Todhunter's  "  Treatise  on  the  DitFer- 
ential  Calculus."     (See  p.  325,  &c.,  of  that  work.) 

1.  Points  cT arret  are  those  points  of  a  curve  at  which  a 
single  l)ranch  of  it  suddenly  stops. 

Thus,  y  =  a?  log  X  =  log  x  -. — 

X 

dx  dx       ^        1 

gives  y:x:_.    _^    __  =   l^_::^.^; 

which  shows  that  y  =  0  when  x  =  0,  or  the  curve  stops 
when  a?  =  0,  which  is  hence  a  point  d' arret/  but  if  a?  is 
negative,  then,  since  log  x  is  impossible,  it  follows  that  y 
must  be  impossible.  For  the  first  part  of  what  has  been  done, 
see  p.  67. 

2.  A  poiiit  saillant  is  a  point  at  which  two  branches  of  a 
curve  meet  and  stop,  without  having  a  common  tangent. 

Thus,  let  y  = j ,  which  gives 

1  +  ^ 

dl  _       1  e^ 

dx  •'  /  i\-* 

1  -^  e"       x\l  +  e^j 

in  which  e  denotes  the  base  of  hyperbolic  logarithms. 


APPENDIX  603 

If  X  is  unlimitedlj  small,  tlien  y  in  the  proposed  equation 
is  unlimitedlj  small  also,  for  two  reasons :  first,  on  account 
of  tlie  smallness  of  x ;  and  second,  on  account  of  the  unlim- 

1  .  .  '- 

itedly  great  value  of  -  in  the  denominator  1  -{-  e"";  and  it  is 

clear  that  the  curve  touches  the  axis  of  x  at  the  origin  of 
the  co-ordinates,  where  y-=  0.  Again,  if  x  is  negative,  it  is 
easy  to  see  that  x  unlimitedlj  small  gives  y  unlimitedlj 
small  at  the  origin  of  the  co-ordinates,  or  where  a?  =  0 ;  and 

it  is  also  clear  that  when  a?  =  0,  we  shall  reduce  —  to 

ax 

|=.^_^  +  &e.=       l        +&,  =  !, 

1  4-  €    ^-  I  +  ex 

so  that  -^  is  the  tangent  of  an  arc  of  45°  ;  and  of  course  the 

second  branch  of  the  curve  lies  on  the  negative  side  of  the 
axis  of  cc,  and  tnakes  an  angle  with  x  negative  of  45°,  or  half 
a  right  angle,  and  intersects  the  preceding  branch  of  the 
curve  at  the  origin  of  the  co-ordinates,  making  an  angle  of 
135°  with  it 

3.  If  a  curve  has  an  infinite  number  of  conjugate  points, 
that  series  of  points  is  called  a  branehe  jpoiiitill&e.  Thus,  if 
y^  =z  X  sin^  a?,  or  y  —  sin  x  ^x,  then,  if  a?  =  -  in  an  j  in- 
tegral multiple  of  tt,  we  shall,  for  all  such  points,  have  y  —  0, 
as  required. 

Bemarks. — Since  these  points  do  not  have  place  in  al- 
gebraic curves,  jet,  since  thej  maj  sometimes  occur  in  tran- 
scendental curves,  we  have  deemed  it  right  to  give  an  account 
of  them. 

II. 

We  here  propose  to  investigate  the  path  that  ought  to  be 
described  bj  a  boat  in  crossing  a  river  of  given  breadth. 


604  APPENDIX. 

fix>m  a  given  point  on  one  side  to  a  given  point  on  the  other, 
so  as  to  make  the  passage  in  the  least  time  possible ;  sup- 
posing the  simple  velocity  of  the  boat  by  the  propelling 
power  to  be  given,  and  that  the  velocity  of  the  current,  being 
in  the  same  direction  with  the  parallel  sides  of  the  river,  is 
variable,  and  expressed  by  any  given  function  of  the  perpen- 
dicular distance  from  that  side  of  the  river  fi-ora  which  the 
boat  sets  out. 

It  is  manifest  that  the  boat,  by  the  propelling  power  alone, 
■will  describe  a  certain  line,  either  straight  or  curved,  passing 
from  her  point  of  departure  to  the  other  side  of  the  river, 
which  is  such  that  the  current  will  float  her  dbwn  the  river 
into  another  curve,  which  is  formed  by  the  composition  of 
the  velocity  of  the  boat  in  the  direction  of  the  first  curve 
and  of  the  velocity  of  the  current,  and  that  the  curve  thus 
described,  from  the  point  of  departure  to  the  point  of  arrival, 
will  be  described  in  the  same  time  that  the  pi'opelling  power 
alone  would  cause  her  to  describe  the  first  curve  mentioned, 
which  time,  by  the  question,  is  to  be  a  minimum. 

Let  then  y,  y',  denote  corresponding  ordinates  of  the  two 
curves  (y  belonging  to  the  first  curve),  having  x  fdr  their 
common  abscissa,  the  origin  of  the  co-ordinates  being  at  the 
point  of  departure,  the  perpendicular  width  of  the  river  being 
the  line  of  the  abscissas,  and  its  side  the  line  of  the  ordinates. 

Let  V  denote  the  given  velocity  of  the  propelling  powei-, 
and  t  the  time  elapsed  from  the  instant  of  departure ;  also, 
let  <px  vary  as  the  velocity  of  the  current  at  the  distance  Xy 
and  let  acftx,  a  being  a  constant,  and  for  simplicity  put 
a(px  =  x'.     Now,  we  have 


^/dx'  +  dy-  =  Vf/^,      .-.  dy-  ^  Y'df^  —  dx\ 
also  dy'  —  dy  =.  x'dt     for     dy'  —  dy 


APPENDIX.  605 

maaifestly  denotes   the   infinitely   small   distance   througli 
which,  the  current  floats  the  boat  in  the  time  dt^ 

:,  chf  -^  {x'dt  -  dyj  =  x'H€'-  "Ix'dtdij' -\-df^  =  YHf"  -  dx'', 

since  dif  =  Yhif  —  dx% 

and  thence    ar  +  ^^7-. — ^  dt  =  -==^ ^ . 

\-^  -~  x^         y-  —  x'^ 

Solving  this  quadratic,  we  have 


_  VY-W'  +  ( V  -  i^")  dz'  -  x'dy' 

^^-  Y'-x'^  ' 

di/ 
which,  by  putting  -^^  =  p,  taking  the  integral,  &c.,  gives 

which  is  to  be  taken  from  the  given  time  of  departure  to 
the  given  point  of  arrival.     Since  Hs  to  be  a  minimum,  its 


1               r^    /VYY  +  Y'  -  x''  -  x'p\ 
equal  j  dx  [^       ^     Y'^  -  x"' J     ' 

must  be  a  minimum  also. 

Since,  by  regarding  dx  as  being  constant  (because  ccMs  a 
fimction  of  x  only),  we  must,  in  taking  the  variations,  con- 
sider x'  as  being  constant,  and  take  the  variations  by  regard- 
ing j9  only  as  variable;  which  gives 

.    r.    (\^Yy  +  Y'-x"-x'p\ 

=  rj^!^^i J!p_ ^\ 

J  Y'-  x"-  \(Yy  +  Y--  _  ^-~f  J 

r    d6y'      /  Vy  ^    A 


606  APPENDIX. 

since  dx^p  =  dd(/'  =  d6i/\ 

by  the  principles  of  the  Calculus  of  Variations.  Integrating 
by  parts,  we  have 

f'^y"'  -  W  -  f^y'd  (yi^.)  X 

by  the  nature  of  maxima  and  minima.  In  which  <^'  and  </> 
correspond  to  the  first  and  last  points  of  the  curve,  which 
being  given,  6y'"  and  6y'\  their  multipliers,  must  be  equal 
to  naught ;  and  of  course  the  preceding  integral  is  reduced 

to    -  hy'd  L,  ^      \  ( ^'^-^ -  x\  =  0, 

which  clearly  can  not  be  satisfied  so  as  to  leave  Sy^  arbitrary, 
except  by  putting  its  factor  equal  to  naught,  which  gives 

whose  integral  gives 


by  using  ^  for  the  arbitraiy  constant. 

This  equation  is  clearly  equivalent  to  the  form 


cvy-  cy  VYy  +  Y'-x'^  =  (V'-x'')  vYy  +  y'-  x'\ 

or  we  shall  have 


CV^  =  (Y^  +  Ox'  -  x'^  VY-y  +  V^  -  x'\ 

or       CVy  =  (V^  -i-  C^'  -  x'J  (Vy  +  Y^  -  x'% 
which  gives 


whicli  gives 


APPENDIX.  607 


^  ~  dx  ~  (Y^j^QY  +  Ox'-x'^^fiGY-Y^-Cx'^-x'^f  ' 


or  we  have 


y(¥_    (Y^  +  C^^  -  x")  VY'  -  x'-'    ^ 
dx  ~  [-Q^ys  _  (y2  ^  c^'  _  aJ'2^^2■]i' 

whose  integral  will  give  the  curve  described  bj  the  pro- 
pelling power  and  the  action  of  the  water  upon  the  boat 
during  its  motion. 

di/ 
To  make  -j-  in  the  preceding  question  real,  the  expression 

(XX 

in  its  denominator  positive,  so  that  the  square  root  can  be 
taken,  and  thence  give  a  real  result. 

If  we  omit  the  terms  in  the  same  expression  that  contain 

^y  ^..  ^y' 


X 


'  and  its  powers,  and  put  -j-  for  -~-^  we  shall,  by  a  simple 

reduction,  s^et     -4-  =  ,     for  the  line  described  by 

'  ^        dx       ^Q^  _  1 '  -^ 

propelling  power  alone,  from  which  the  current  may  be  sup 

posed  to  float  the  boat  down  into  the  actual  curve  described, 

has  the  preceding  for  its  differential  equation.     Because  Y 

and  C  are  invariable,  it  is  clear  that  the  integral  of  the  pre 

Yx 
ceding  equation  is  y  =  —- — — ,  which  needs  no  correction, 

supposing  the  origin  of  the  co-ordinates  to  be  at  the  place  of 
departure  of  the  boat 


608  APPENDIX. 

Hence,  the  propelling  power  alone  causes  the  boat  to 
describe  a  right  line  passing  through  the  given  place  of 
departure.  To  get  C,  we  must  obtain  the  integral  of  the 
preceding  equation,  in  which,  bj  putting  for  y'  its  value  at 
the  given  point  of  arrival,  noticing  that  the  correction  may 
clearly  be  supposed  to  equal  naught,  we  shall  have  an 
equation  whose  only  unknown  quantity  will  be  C,  which 
solved  gives  C ;  and  thence,  by  taking  those  values  that  are 
not  less  than  1,  the  first  and  last  points  of  the  right  line, 
described  by  the  propelling  power  alone  become  known, 
and  thence  the  direction  or  directions,  according  as  C  has 
one  or  more  values,  will  be  found,  and  the  problem  solved, 
as  required. 

Remarks. — The  question  here  solved  was  proposed  in 
No.  2  of  the  "Mathematical  Diary,"  in  the  year  1825,  by 
its  much  accomplished  editor  and  profound  mathematician, 
Robert  Adrain,  LL..D.,  then  professor  of  mathematics  in 
Columbia  College,  New  York.  I  communicated  a  solution 
to  the  question  in  No.  3  of  the  same  work,  which  received 
the  prize  awarded  the  solution  by  the  editor.  Since  there 
were  many  mistakes  in  the  published  solution,  I  have  con- 
cluded, at  the  earnest  solicitation  of  a  former  pupil  and  a 
much  accomplished  scientific  gentleman,  to  insert  the  correct 
solution  of  the  question  in  tliis  work. 

III. 

To  illustrate  what  has  been  done,  suppose  the  body  A 
moves  uniformly  around  the  circumference  of  the  circle 
LKI  with  a  velocity  represented  by  1,  or  unity,  while  the 
body  B,  in  pursuit  of  A,  moves  continually  directly  toward 
A  in  the  curve  BB'B''  witb  the  uniform  velocity  ?/* ;  then 


APPENDIX. 


609 


it  is  proposed  to  show  bow  to  find  the  nature  of  the  curve 
described  by  B,  or  of  the  curve  of  pursuit. 

Let  A  A'  and  BB'  be  very  small  parts  of  the  curves 
described  by  A  and  B  in  the  same  time,  and  they  will 
clearly  have  the  ratio  1  :  m..  Let  O  be  the  center  of  the 
circle  connected  with  the   extremities  of  the  arc  A  A'  by 


5:0 


the  radii  AO,  A'O,  at  whose  extremities  the  tangents  Aa 
and  A! a"  are  drawn,  crossing  each  other  in  C ;  then  it  is 
evident  we  shall  have  the  angle  aC'A',  made  by  the  tangents, 
equal  to  the  angle  A'OA,  subtended  by  the  arc  at  the  center 
of  the  circle.     Now  the  angle. 

aAB  —  (p  =z  Aa'h  +  A'JA, 
AB   and   A'B'   being    the  corresponding   tangents   of    the 
curve  of  pursuit  which  intersect  in  ^,  and  thence  we  have 
A'hA  =  aAB  —  Aa'b  —  (p  —  Aa'h 

=  cp  -  (CA'5  -  AVa')  =  -  d<p  -[-  AOA'     " 

by  the  nature  of  a  differential,  since  the  angle  a'^A'B'  (sup- 
posed to  decrease)  is  successive  to  aAB  =  0.  Putting 
AO  =  r  and  the  arc  AA'  =  dx^  we  have 


A'hAr=z 


^0  +  ^; 


GIO  APPENDIX. 

and  thence,  from  the  tritmgle, 

A'hA  sin  (—  -  dtjA  = 

dx 

d(p   :  dx  :\  sin  AA'^     or    0  (ultimately')  :  Ah  =  t, 

which  gives  -  dx  —  td(f)  =  sin  (pdx. 

Drawing  Ae  perpendicular  to  A'^,  we  have  A'e  =  cos  ff>dx, 
and  ultimately  eb  =  Ai,  or  eWB  =  A^B  ;  and  tbence 

~  d£  =z  B'B  —  A'<?  =  —  cos  (fxlx  -f  7ndxy 

which  gives  dx  :^. . 

m  —  cos  (j) 

From  the  substitution  of  the  value  of  dx,  the  preceding 
equation  reduces  to 

tdt       ,  .  -  .        ,         / 
-~  {rn  —  cos  (p)  —  td^  =  —  sm  (pdt  -i-  (m  —  cos  (p), 

or  tdl  =  r  \d  (sin  <^,  t)  —  mid<f)^. 

To  integrate  this  equation,  we  may  clearly  assume 

t  =  A0  H-  B0-^  -f  C9'  +,  &c., 
which  gives 

Ut  =  ^  [A</>  +  B0^  -f  &c.]^ 
z 

=  AV<^,^  +  4ABvVZ/)  +  3  (2 AC  +  B-)  cpMcp  +,  <?cc.  ; 
also  from 

^       ^       1.2.8^1.2.3.4.5      '^^•' 
we  easily  get  d  (sin  0, 25)  = 

2A  .c^H-  (-  I  A  +  4B)  0VZ^  +  ( -^  _  I  +  c)  6<A-V0  +,  &c., 

and     —  midxp  =  —  mA6d4>  —  mB(l>\I<p  —  7nCfd(l>  — ,  &c. 


APPENDIX.  611 

Hence,  from  substitution  and  omitting  the  factor  c?0,  the 
equation  tdt  —  r  [d  (sin  0,  t)  —  vitdcp] 

becomes  the  identical  equation 

A^*/)  +  4AB</>^  +  3  (2  AC  +  B^)  cf>^  +  &c.  = 

(2A  —  mA)  r^  +  (—  •-  A  -f  4:B  —  7nB\  r(p^  + 

(^  -  B  -f  60  -  mC)  a^</)^  +,  &c.  ; 

which,  by  equating  the  coefficients  of  like  powers  of  0,  gives 

A  =  (2  -  7n)  r,     4AB  =  ^-  |  A  +  4B  -  m^\  /•, 

or  (4  -  3w)  B  =  -  I  A, 

o 


which  gives  B  =  -3(-^— |i)., 

and        3  (2  AC  +  B=)  ==  /^  -  B  +  6C  -  ??iCW ; 

or  we  have 

6AC  -  6C/'  -f  /r.Cr  =  /A  _  b)  /•  -  SB"^ 

_  (2  -  m)  (52  -  9?7^)   ^  _ 
~        60(T-8w.)        ^*'' 


SB^ 


2 


or    (6  -  5;.)  C  =  (^I(-5?_Zll!rO  ,  _  4  .(^J!!) 
V."       oni)  Kj  60  (4  -  3/7^)  3  (4  -  2>7n) 

_  {2~m)  (4-3??i)  (52  -  9m)  -  80  (2-?/?y 

■~  60^(4 -3/«)-  ~' 

which  gives 

_  (2  -  m)  (4  -  3m)  (52  -  9m)  -  80  (2  -  mf 

60  (4  -  3my  (6  -  5m)  ^' 

and  so  on.     Hence, 


612  APPENDIX. 

(2  -  m)  4  -  3m)  (52  -  9m)  -  80  (2  -  ^nf  . «   ,    „    ]  ^ 
60^-  Smf  (6  -  5m)  ^  +  ^""'j  "^ 

is  the  integral,  Q'  being  the  constant ;  and  if  T  =  C  is  the 
value  of  t  at  the  origin,  when  0  =  </>',  we  shall  clearly  have 

,=.T4-((2-m)(^-,0-i  (,^3^^(0-00^^ 

(2  _  ,n)  (4  -  3m).(52  -  9m)  -  80 (2-m.)^       _  | 

60  (4  -  ^6vif  (6  -  5m)  (^      ^  )  +  &c.  j  / 

for  the  correct  integral. 

To  find  a?,  we  take  the  equation  dx  =^ ;  then, 

^  m  —  cos  0 

from  the  value  of  t^  we  get  the  form 

—  dt  =  —  [AcZ^  +  3B  ((^  -  ^Jd^  4-  5C  ((^  -  (i>yd(l>  H-  &c.]  ; 

consequently,  since 

by  putting  m  —  1  =  m',  we  get 

whose  integral  gives 

(m'  +  6)  A  -  36m'B  +  120m"O  ,^   ,    ^,-.    ,    ,     1 

IJO^^^J —  (*  +  ^ )  +  ^'-l  '•' 

which   needs  no  correction,  supposing  it  commences  with 


APPENDIX. 


613 


(p  =  (p\  or  to  equal  nauglit  at  tlie  origin  of  the  motion.     If 
in  this  value  of  x  we  put 

as  in  tj  we  shall  get  the  required  value  of  x. 

By  taking  (f)  —  (p'  sufficiently  small,  we  can,  from  the  for- 
mulas found,  find  the  corresponding  values  of  t  and  x ;  and 
then,  changing  ^  into  <t>\  and  putting  <p  —  <p'  for  a  new  value 
of  0  —  <^',  we  may  calculate  the  corresponding  values  of  t 
and  a?  as  before,  and  so  on,  to  any  required  extent;  conse- 
quently, in  this  way  we  may  find  any  number  of  points  in 
the  required  curve  of  pursuit. 

Eemark. — Tliis  example  is  given  to  illustrate  the  method 
of  integration  given  by  the  series  in  Remark  1,  at  p.  555. 

On  account  of  the  complication  of  the  preceding  process, 
we  will  insert  a  more  simple  method  by  linear  description. 


Thus,  let  A  start  from  the  position  1  in  the  circumference 
of  the  circle,  while  B  starts  from  0  within  the  circle,  and 
let  the  bodies  move  uniformly  over  the  distances  1,  2  ;  2,  3  ; 


614  APPENDIX. 

3,4,  &c,«and  over  the  distances  0,1;  1,2;  2,3,  &;c.,  such, 
that  the  first  distances  being  each  represented  by  1,  the 
second  distances  (01,  &c.)  shall  each  be  represented  by  m. 
Then  will  the  curve  described  by  B,  the  pursuing  body,  be 
represented  by  the  rectilineal  figure  0,  1,  2,  3,  &c.,  nearly, 
and  thence  the  curve  of  pursuit  can  be  approximately  found ; 
and  it  is  evident  that  the  figure  can  be  described  in  a  slightly 
different  manner,  which  sensibly  gives  the  same  result  as 
before. 

IV. 

Having  procured  the  last  work  on  Quaternions  by  the 
late  Sir  William  Rowan  Hamilton,  hL.J).^  M.RI.A.,  &c. 
published  in  1866,  since  the  much  lamented  death  of  the 
gifted  author,  by  his  son  William  Edwin  Hamilton,  and 
having  given  much  time  to  the  study  of  the  work,  we  here 
propose  to  notice  some  parts  of  the  work  that  are  somewhat 
analogous  to  what  has  been  done  in  the  preceding  treatise. 

To  the  end  in  view,  we  shall  refer  to  what  is  done  by  the 
author  at  p.  215  of  his  treatise,  as  follows: 

(1.)  From  a  point  A  of  a  sphere  with  0  for  center,  let  it  be 
required  to  draw  a  chord  AP,  which  shall  be  parallel  to  a 
given  line  OB;  or  more  fully,  to  assign  the  vector^  q  =  OP, 
of  the  extremity  of  the  chord  so  drawn,  as  a  function  of 
the  two  given  vectors,  a  ==  0 A  and  (3  ==  OB ;  or  rather,  of 
a  and  UB,  since  it  is  evident  that  the  length  of  the  line  /3 
can  not  affect  the  result  of  the  construction,  which  the  figure 
may  serve  to  illustrate. 

(2.)  Since  AP  1  OB  or  p  —  o  [  i3,  we  may  begin  by 
writing  the  expression 

9  =  a  +  osp (1), 


APPENDIX. 


615 


which  may  be  considered  as  a  form  of  the  equation  of  the 

right  line  AP,   and  in  which  it  remains  to  determine  the 

scalar  coefficient  a?,  so  as  to  satisfy  the  equation  of  the  sphere 

Tp  =  Ta (2). 


In  short,  we  are  to  seek  to  satisfy  the  equation 

T  (a  +  x^)  =  Ta 
by  some  scalar  x  which  shall  be  in  general  different  from 
zero,  and  then  to  substitute  this  scalar  in  the  expression 
Q  =  a  -\-  x(i,  in  order  to  determine  the  required  vector  \ 

(3.)  For  this  purpose,  an  obvious  process  is,  after  dividing- 
by  T/3,  to  square,  and  to  employ  the  formula  210,  XXL, 
which  had  indeed  occurred  before,  as  200,  YIIT.,  but  not 
then  as  a  consequence  of  the  distributive  property  of  multi- 
plication.    In  this  way  we  get 


\Tfi)   ~  (t/3  +  V  ' 


2a 

or     -J--  X  -\-  X- 


0, 


which  is  satisfied  either  by 


which  "fives  x 


2a 
0,     or     -Q    +  X 

2a 

T 


Substituting  this  value  for  x  in 


616  APPENDIX. 

the  equation  q  =  a  -\-  I3x,  we  have  p  =  a  —  2a=z  —  a,  on 
account  of  the  negative  sign ;  consequently,  AP  is  clearly 
parallel  to  OB  as  required,  or,  as  in  Hamilton, 

CA  =  -^  =  a, 

and  the  figure  OC AD  is  a  parallelogram. 

Instead  of  this  process,  we  raise  the  members  of  the  equa- 
tion ^  =  a?  +  73  to  the  integral  power  n,  and  retain  only 

those  terms  that  involve  the  first  power  of  i3,  and  thence,  on 
account  of  the  indefiniteness  of  [3,  we  get 

0  =  2?**  +  wa?"-^  ^, 

which  gives  x  = ^ ;  consequently,  from  p  =  a  -f  a?/?,  we 

have  p  =  —  (ti  —  1)  a,  which,  for  n  =  2,  gives  p  =  —  a,  as 
before.  It  is  evident  that  we  may  in  like  manner  take 
n  =  3,  4,  5,  &c.,  and  thence  draw  any  number  of  parallels  to 
OB,  which  will  not,  however,  pass  through  the  point  A; 
and  it  is  evident  that,  in  like  manner,  a  parallel  to  OB  may 
be  drawn  through  E,  as  in  the  figure.  Thus  a  sphere,  having 
its  center  at  0  and  radius  OE,  cuts  PT  produced  toward 
P',  so  that  its  arc  between  E  and  where  it  meets  OP'  pro- 
duced will  be  bisected  by  OB ;  and  of  course  the  right  line 
EP°P',  joining  E,  and  where  OP  produced  meets  the  sphere, 
must  be  parallel  to  OB. 

Remarks. — It  is  clear  that  the  preceding  method  of  treat- 
ing the  problem  will  be  found  useful  in  all  analogous  cases. 
We  will  add  that  Hamilton  represents  a  quaternion  in  its 
most  general  form  hj  q  =  (^  i-  ix  +/'/  +  /<^2,  in  which  the 
term  w,  and  the  multipliers  x,  y,  z,  are  called  scalars, 


APPENDIX.  617 

^^  =:y-  =:  ^-  z=  ijli  =  —   1, 

'/,  j,  /i',  being  so  taken  that  they  shall  represent  a  system  of 
three  right  versors  in  three  rectangular  planes,  as  described 
by  Hamilton  at  p.  157,  Art  181 ;  and  for  the  method  of 
using  these  versors  in  practice  we  shall  refer  to  p.  366,  &c., 
of  Hamilton,  where  he  will  find  some  well-known  formulas 
of  spherical  trigonometry. 

Finally,  we  would  advise  the  student  to  make  himself 
familiar  with  Hamilton's  definitions  ;  to  read  with  care  the 
parts  of  the  work  to  which  reference  has  been  made.  Be- 
sides he  will  also  do  well  to  read  from  p.  208,  Art.  210,  con- 
tinuously to  p.  240,  Chapter  II.  Indeed,  whoever  will 
give  his  attention  to  obtain  a  thorough  knowledge  of  the 
work,  vfill  find  his  labor  abundantly  rewarded. 


THE  ENI>. 


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